Geometric Relation Algebra

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The Geometric Relation Algebra , founded by Hans-Joachim Arnold , is a specialization of Geometric Algebra and thus a sub-area of incidence or synthetic geometry on the one hand and universal algebra on the other . Based on the objective of geometric algebra, it is very easy to calculate with objects of geometrical perception such as planes, angles and circles as well as with geometric operations such as sections of different objects or transformations and essentially focus on the geometric interpretation of algebraic systems such as vector algebras , quaternions, etc. . to use, in the geometric relation algebra (almost) exclusively relation systems for the algebraic description of geometric artifacts are used. This calculus, which has been developed since the mid-1970s, for the first time completely answers the question of a synonymous, i.e. isomorphic - and not as before with the help of ternary bodies and other models only isotopic - algebraic description of all affine and projective geometries and is made possible by the principle of "operator operandum" an intuitively simple access to a remote space geometry . The most important aspect is that for geometrical intersection point theorems (e.g. affine and projective theorem of Desargues , invariance theorem of Hessenberg ) in the geometric relational algebra necessary and sufficient conditions with simple "calculation rules", i.e. the sequential execution of relations in the " Homogeneity rules ", specify and have proven (" principle of constructive expandability "). The methodology is also used in the cognitive sciences and especially in systems theory : Here, the relational calculus describes systems as different as time-discrete, time-continuous (linear and non-linear) dynamic systems and fuzzy systems, including important system properties, using the same mathematical language (synonymously) . Central terms in geometric relational algebra are geometric relatives , rule relatives and action relatives - and the derivatives derived from them.

methodology

In geometric relational algebra, geometric configurations such as straight lines, angles, triangles are represented as systems of relations of a suitable relational algebra , which are applied to individual points of a point set. For example, a straight line in an affine geometry with a set of points in the synonymous description of an affine relative, i.e. a set of two-digit relations , can be represented as for a given one that represents an equivalence class of parallel straight lines. can then be understood to mean that the straight line runs through and in this class, see also the transition procedure in 2.1.1.1. With this approach it was possible to finally solve the central concern of geometric algebra, a synonymous algebraic characterization of all affine and projective geometries, with concepts from relational algebra. Showed in this issue of the last century to date provided structures of universal algebra as vector spaces over skew fields , planar ternary ring , quasi modules etc. due to lack of coordinate areas or because of dependencies on the choice of the transition process required coordinate system glaring weaknesses: So are those of Marshall Hall introduced ternary body for the description of any projective planes unsatisfactory, since the identification i. A. is not unique, but only isotopic , d. that is, one and the same projective plane can induce two non- isomorphic ternary bodies. The quasi-modules introduced by Emanuel Sperner also have the same shortcomings. A further complication is that in some algebraizations the validity of Desargues' theorem must be assumed. Since the construction of the Moulton plane , non-Desaguean affine planes have even been known, although every affine space is desarguic. ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle A{\mathfrak {B}}=\{C\in {\mathfrak {P}}\mid A{\mathfrak {B}}C\}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle A{\mathfrak {B}}C}$ ${\displaystyle A}$ ${\displaystyle C}$

With the two-digit affine and projective relatives developed by Hans-Joachim Arnold in the 1970s (the latter were initially referred to by him as projective multigroups or projective point operations based on earlier names by Walter Prenowitz), which consist of a set of relations, which operate on the set of points of the submitted geometry, reverse the transition processes of algebraization and geometrization synonymously, i.e. H. except for isomorphism, around each other. In a further step, the elements of the set of relations that have an operational effect on the basic set themselves also become objects of secondary operation by means of the product of relations. The principle of "operator operandum" of algebraization expressed in this way finds its direct application in the derivation of the projective multigroups, which serve the algebraic description of the projective distance-space structure of the underlying geometry. Another advantage of the relational algebraic methodology lies in its "constructive expandability": Without having to leave the selected relational algebra language, this calculus is suitable for specifying simple and easy-to-use calculation rules that are equivalent to extensive additional geometrical axioms (closing sentences). In particular, the question of geometric algebra that has been raised since Hilbert's Fundamentals of Geometry , whether a given geometry is desarguean, is answered on the basis of its now existing algebraic equivalent with the validity of a certain rule of linking the underlying relations - the affine or projective homogeneity rule. The calculations of universal algebra that have been used up to now only provide descriptions of the geometrical statements that do not go beyond the geometrical-combinatorial work with intersection configurations.

The addition of a directional term to the affine relative is not only an adequate description of arranged geometries, but also provides insights into cognitive theory : The concept of the action scheme, coined by the developmental psychologists Jean Piaget and Hans Aebli , is interpreted with binary fields of action, which in turn can be described as their relational Are algebraization by affine directional relatives.

The language of geometric relational algebra has proven to be so powerful that, in addition to the established differential algebraic or differential geometric calculi, it provides another method for the (synonymous) description of any linear , nonlinear and fuzzy systems of control theory and is also able to use geometric systems for the first time To indicate closure sentences in the state spaces of the dynamic systems with the corresponding relational system calculus.

Basic terms and results

Synthetic geometry

Synonyms context

Affine part

One speaks of an n-digit relative with if the following are given: ${\displaystyle ({\mathfrak {P}},{\mathcal {R}}^{(n)})}$ ${\displaystyle n\geq 2}$

• a lot of points ${\displaystyle {\mathfrak {P}}=\{A,B,C\dots \}\neq \emptyset .}$
• a set of -digit relations on the basic set ${\displaystyle {\mathcal {R}}^{(n)}=\{{\mathfrak {A,B,C,}}\dots \}\subset \mathrm {Pot} ({\mathfrak {P}}^{(n)})\setminus \left\{\emptyset \right\}}$ ${\displaystyle n}$ ${\displaystyle {\mathfrak {P}}.}$

A two-digit relative is simply called graphical if it fulfills the following properties: ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$

1. Scharf simple transitivity : . There is exactly one "connection" relation for two given points , one sets: ${\displaystyle \bigwedge _{A,B\in {\mathfrak {P}}}\bigvee _{{\mathfrak {D}}\in {\mathcal {R}}}^{1}:A{\mathfrak {D}}B}$ ${\displaystyle A,B\in {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {D}}\in {\mathcal {R}}}$ ${\displaystyle \;}$ ${\displaystyle {\mathfrak {D}}=AB:\Leftrightarrow A{\mathfrak {D}}B.}$
2. Seclusion regarding equality relation : . Equivalent to this is the fact that for the equality relation on applies: . ${\displaystyle \bigwedge _{A,B,C\in {\mathfrak {P}}}:A(BB)C\Leftrightarrow A=C}$ ${\displaystyle {\mathfrak {E}}}$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {E}}\in {\mathcal {R}}}$
3. Seclusion regarding inverse : . The inverse relation (inverse relation) to a given relation is also contained in. ${\displaystyle \bigwedge _{A,B\in {\mathfrak {P}}}:AB^{-1}=BA}$ ${\displaystyle {\mathfrak {D}}^{-1}}$ ${\displaystyle {\mathfrak {D}}}$ ${\displaystyle {\mathcal {R}}}$
4. Links totality : . The relations suffice for a certain richness. ${\displaystyle \bigwedge _{A\in {\mathfrak {P}}}\bigwedge _{{\mathfrak {D}}\in {\mathcal {R}}}\bigvee _{B\in {\mathfrak {P}}}:A{\mathfrak {D}}B}$

A two-digit, simply graphical relative is called homogeneous if: ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$

${\displaystyle (H_{2})\bigwedge _{A,B\in {\mathfrak {P}}}\bigwedge _{{\mathfrak {A,B}}\in {\mathcal {R}}}:A({\mathfrak {A}}\circ {\mathfrak {B}})B\Rightarrow AB\subset {\mathfrak {A}}\circ {\mathfrak {B}}.}$

This formula is called the two-step homogeneity rule, it is equivalent to the following expression:

${\displaystyle (H_{2})\bigwedge _{A,B,C\in {\mathfrak {P}}}:AB\subset AC\circ CB.}$

A two-digit, simply graphical and homogeneous relative is called an affine directional relative if the relations satisfy the following properties: ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$

1. Strictly alternating relations : ${\displaystyle {\mathfrak {A}}\circ {\mathfrak {A}}^{-1}={\mathfrak {A}}\cup {\mathfrak {A}}^{-1}\cup {\mathfrak {E}}}$
2. Idempotent relations : ${\displaystyle {\mathfrak {A}}\circ {\mathfrak {A}}={\mathfrak {A}}}$
3. Antisymmetric relations : ${\displaystyle {\mathfrak {A}}\cap {\mathfrak {A}}^{-1}=\emptyset }$
4. Commutating relations : ${\displaystyle {\mathfrak {A}}\circ {\mathfrak {B}}={\mathfrak {B}}\circ {\mathfrak {A}}}$

One speaks of an affine relative if the relative is simply graphical and homogeneous and the relations are symmetrically alternating , that is: ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$ ${\displaystyle {\mathcal {R}}}$

1. Symmetry : ${\displaystyle {\mathfrak {A}}={\mathfrak {A}}^{-1}}$
2. Alternating relations : ${\displaystyle {\mathfrak {A}}\circ {\mathfrak {A}}^{-1}\subset {\mathfrak {A}}\cup {\mathfrak {E}}}$

Every affine directional relative creates an affine relative if it is set: ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$ ${\displaystyle ({\mathfrak {P}},{\mathcal {\tilde {R}}})}$ ${\displaystyle {\mathcal {\tilde {R}}}:=\{{\mathfrak {A}}\cup {\mathfrak {A}}^{-1}\mid {\mathfrak {A}}\in {\mathcal {R}}\}.}$

One speaks of an affine geometry if the following are given: ${\displaystyle ({\mathfrak {P}},{\mathfrak {G}},\in ,\parallel )}$

• a lot of points ${\displaystyle {\mathfrak {P}}=\{A,B,C,\dots \}\neq \emptyset ,}$
• a lot of straight lines ${\displaystyle {\mathfrak {G}}=\{g,h,k,\dots \}\subset \mathrm {Pot} ({\mathfrak {P}}),}$
• the set theoretical element relation as incidence relation, ${\displaystyle \in }$
• a parallel relation ${\displaystyle \parallel }$ ${\displaystyle {\mathfrak {G}}}$

and if the following properties apply:

1. Existence and uniqueness of the connecting line: Through two different points is exactly one line with . It is for this uniquely determined connecting line also . ${\displaystyle A,\,B}$ ${\displaystyle g}$ ${\displaystyle A,B\in g}$ ${\displaystyle g}$ ${\displaystyle g_{AB}}$
2. Lines are connecting lines: There are at least two points on every line.
3. The parallel relation is an equivalence relation ${\displaystyle \parallel }$
4. Euclidean postulate of parallels: Exactly one straight line passes through each point and is parallel to a given straight line.
5. Constructibility of parallel-like triangles ( Tamaschke axiom ): If a triangle (three points not lying on a straight line) is given and two points and such that the straight line is parallel to the straight line , there is a point such that also parallel to and to lie parallel . ${\displaystyle ABC}$ ${\displaystyle A'}$ ${\displaystyle B'}$ ${\displaystyle AB}$ ${\displaystyle A'B'}$ ${\displaystyle C'}$ ${\displaystyle AC}$ ${\displaystyle A'C'}$ ${\displaystyle BC}$ ${\displaystyle B'C'}$

An affine geometry is created from an affine relative using the following procedure ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$ ${\displaystyle \gamma }$ ${\displaystyle ({\mathfrak {P}},{\mathfrak {G}},\in ,\parallel )=({\mathfrak {P}},{\mathcal {R}})\gamma :}$

${\displaystyle {\mathfrak {G}}:=\{A({\mathfrak {B}}\cup {\mathfrak {E}})\mid A\in {\mathfrak {P}}\wedge {\mathfrak {B}}\in {\mathcal {R}}\setminus \{{\mathfrak {E}}\}\}}$

${\displaystyle A({\mathfrak {B}}\cup {\mathfrak {E}})\parallel A'({\mathfrak {B}}'\cup {\mathfrak {E}}):\Leftrightarrow {\mathfrak {B}}={\mathfrak {B}}'.}$

An affine relative arises from an affine geometry by applying the procedure ${\displaystyle ({\mathfrak {P}},{\mathfrak {G}},\in ,\parallel )}$ ${\displaystyle \alpha }$ ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})=({\mathfrak {P}},{\mathfrak {G}},\in ,\parallel )\alpha :}$

${\displaystyle {\mathcal {R}}:=\{\left\langle g\right\rangle \subset {\mathfrak {P}}^{2}\mid g\in {\mathfrak {G}}\}\cup \{{\mathfrak {E}}\}}$

${\displaystyle A\left\langle g\right\rangle B:\Leftrightarrow A\neq B\wedge g_{A,B}\parallel g.}$

For all affine relatives and all affine geometries : ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$ ${\displaystyle ({\mathfrak {P}},{\mathfrak {G}},\in ,\parallel )}$

${\displaystyle ({\mathfrak {P}},{\mathfrak {G}},\in ,\parallel )\alpha \gamma \cong ({\mathfrak {P}},{\mathfrak {G}},\in ,\parallel ).}$

${\displaystyle ({\mathfrak {P}},{\mathcal {R}})\gamma \alpha \cong ({\mathfrak {P}},{\mathcal {R}}).}$

Affine relative and affine geometries are synonymous with one another, the (H2) -homogeneity rule on the algebraic side corresponds to the constructibility of parallel-like triangles on the geometric side.

The class of three-digit affine relatives also turns out to be synonymous with the class of affine geometries. A two-digit, simply graphic and homogeneous relative , i.e. a relative whose set of relations is sharply, simply transitive, closed with respect to equality relation and inversion, left total and homogeneous, is also defined ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$

${\displaystyle {\begin{cases}{\mathcal {R}}^{(3)}\;:=\;\{ABC\mid A,B,C\in {\mathfrak {P}}\}\\(A',B',C')\in ABC\;:\Leftrightarrow \;A'(AB)B'\wedge B'(BC)C'\wedge A'(AC)C'\end{cases}}}$

a simple graphic three-digit relative . ${\displaystyle ({\mathfrak {P}},{\mathcal {R}}^{(3)})}$

With the definition using a projection functor on 2 places each of the three-digit relations, a synonymous connection is established on the one hand between the class of two-digit simple graphic relatives, which are moreover homogeneous, and on the other hand the class of three-digit simple graphic relatives. These are called affine if their 2-digit projection is. ${\displaystyle ({\mathfrak {P}},{\mathcal {R}}^{(3)})\pi :=({\mathfrak {P}},{\mathcal {R}}^{(3)}{\mathcal {F}}_{2})}$ ${\displaystyle {\mathcal {F}}_{2}}$

Projective part

One speaks of a multigroup when the following are given: ${\displaystyle ({\mathcal {P}};\cdot ,{\overline {.}},{\mathfrak {E}})}$

• a lot ${\displaystyle {\mathcal {P}}=\{{\mathfrak {A,B,C,}}\dots \}}$
• a two-digit operation

${\displaystyle \left\{{\begin{array}{c}{\mathcal {P}}\times {\mathcal {P}}\longrightarrow \mathrm {Pot} \;{\mathcal {P}}\setminus \{\emptyset \}\\{\mathfrak {A,B}}\longmapsto {\mathfrak {A}}{\mathfrak {B}}:={\mathfrak {A}}\cdot {\mathfrak {B}}\end{array}}\right.}$

• an involution

${\displaystyle \left\{{\begin{array}{c}{\mathcal {P}}\longrightarrow {\mathcal {P}}\\{\mathfrak {A}}\longmapsto {\overline {\mathfrak {A}}}\end{array}}\right.}$

• a neutral element with ${\displaystyle {\mathfrak {E}}\in {\mathcal {P}}}$ ${\displaystyle {\overline {\mathfrak {E}}}={\mathfrak {E}}}$

and if the following properties with the extension

${\displaystyle \mathrm {Pot} \;{\mathcal {P}}\setminus \{\emptyset \}\ni {\mathcal {A}},{\mathcal {B}}\mapsto {\mathcal {A}}{\mathcal {B}}:=\bigcup _{{\mathfrak {A}}\in {\mathcal {A}},{\mathfrak {B}}\in {\mathcal {B}}}{\mathfrak {A}}{\mathfrak {B}}}$

be valid:

1. Existence of a neutral element: . ${\displaystyle {\mathfrak {A}}{\mathfrak {E}}={\mathfrak {E}}{\mathfrak {A}}=\{{\mathfrak {A}}\}}$
2. Idempotenzregel: . ${\displaystyle {\mathfrak {A}}{\overline {\mathfrak {A}}}\subset \{{\mathfrak {A}},{\overline {\mathfrak {A}}},{\mathfrak {E}}\}}$
3. Exchange rule . ${\displaystyle {\mathfrak {A}}{\mathfrak {B}}\ni {\mathfrak {C}}\;\succ \;{\mathfrak {C}}{\overline {\mathfrak {B}}}\ni {\mathfrak {A}}}$
4. Associative law: . ${\displaystyle ({\mathfrak {A}}{\mathfrak {B}}){\mathfrak {C}}\;=\;{\mathfrak {A}}({\mathfrak {B}}{\mathfrak {C}})}$
5. Commutative: . ${\displaystyle {\overline {{\mathfrak {A}}{\mathfrak {B}}}}\;=\;{\overline {\mathfrak {B}}}\;{\overline {\mathfrak {A}}}}$

${\displaystyle {\mathfrak {A}}{\mathfrak {B}}}$is a product of the multiplication sign quoted in with ; the painting point is no longer written. If the involutorial anti-automorphism of a multigroup is equal to the identity , then one speaks of a projective multigroup . ${\displaystyle ({\mathcal {P}};\cdot ,{\overline {.}},{\mathfrak {E}})}$ ${\displaystyle \;\cdot \;}$ ${\displaystyle \;{\overline {.}}\;}$ ${\displaystyle ({\mathcal {P}};\cdot ,{\overline {.}},{\mathfrak {E}})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle ({\mathcal {P}};\cdot ,{\mathfrak {E}})}$

One speaks of a projective geometry if the following are given: ${\displaystyle (\mathbb {P} ,\mathbb {G} ,\in )}$

• a lot of points ${\displaystyle \mathbb {P} =\{A,B,C,\dots \}\neq \emptyset ,}$
• a lot of straight lines ${\displaystyle \mathbb {G} =\{g,h,k,\dots \}\subset \mathrm {Pot} (\mathbb {P} ),}$
• the set theoretical element relation as incidence relation, ${\displaystyle \in }$

and if the following axioms hold:

1. Existence and uniqueness of the connecting straight: Are and two different points, so there is exactly one line , with and incised . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle g_{AB}}$ ${\displaystyle A}$ ${\displaystyle B}$
2. Abundance axion: There are at least three points on every straight line
3. Veblensches axiom : If , , , , four points so and incident with a common point, so incise also and to a common point. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle g_{A,B}}$ ${\displaystyle g_{C,D}}$ ${\displaystyle g_{A,C}}$ ${\displaystyle g_{B,D}}$

A projective multigroup arises from a projective geometry with the characteristic of its order, i.e. the number of points on a straight line , if one sets: ${\displaystyle (\mathbb {P} ,\mathbb {G} ,\in )}$ ${\displaystyle \lambda }$ ${\displaystyle ({\mathcal {P}}:=\mathbb {P} \cup \{{\mathfrak {E}}\};\cdot ,{\mathfrak {E}})}$ ${\displaystyle \bigwedge _{A,B\in \mathbb {P} }\;A\cdot B:={\begin{cases}\{A\},&{\text{falls }}\;A=B,\,\lambda =3\\\{A,{\mathfrak {E}}\},&{\text{falls }}\;A=B,\,\lambda >3\\g_{A,B},&{\text{falls }}\;A\neq B.\end{cases}}}$

From a projective Multigrppe a projective geometry created with ${\displaystyle ({\mathcal {P}};\cdot ,{\mathfrak {E}})}$ ${\displaystyle (\mathbb {P} :={\mathcal {P}}\setminus \{{\mathfrak {E}}\},\mathbb {G} ,\in )}$ ${\displaystyle \mathbb {G} :=\{{\mathfrak {A}}+{\mathfrak {B}}\mid {\mathfrak {A}}\neq {\mathfrak {B}}\in \mathbb {P} \},\;{\mathfrak {A}}+{\mathfrak {B}}:={\mathfrak {A}}{\mathfrak {B}}\cup \{{\mathfrak {A}},{\mathfrak {B}}\}}$

Due to the synonymous context, projective multigroups and projective geometries are two different ways of speaking for one and the same issue.

More synonymous relationships

With the affine relatives, at that time still without rules of homogeneity, Arnold - a student of Emanuel Sperner - completely and synonymously describes the Sperner Spaces in the 1970s , the counterpart of the projective multigroups / projective relatives, also without rules of homogeneity, then the projective geometries. Each operator of a binary affine relative corresponds to the (still unordered) direction of a parallel set in the associated affine geometry, the linkage of a projective multigroup , written as a product, means the drawing of the connecting straight line in the associated projective geometry. With the addition of an antisymmetry of the operators in the affine relative structures and the introduction of an inversion and the decomposition of a point into a disordered pair of mutually inverse algebra elements in the projective relative / multigroups, Hans-Joachim Arnold then succeeds in decomposing the straight lines into two opposite half-rays analogously the Hilbert interlayer, the result is a synonymous description of Hilbertsch arranged affine and projective geometries by affine direction-relative and projective point operations / multigroups with involution. Thomas Ledabo weakens the axioms of both performing algebras and from such a way that arranged with its affinity preference relative and projective Punktalgebren and not can be arranged in the proper sense models affine and projective geometries, including its remote structures under a uniform Algebraisierungsverfahren and a uniform algebraic structure can be viewed. Orientation functions such as those defined by Helmut Karzel and Hanfried Lenz for Sperner's order functions are used for the synonymous algebraization of Sperner's arranged - in short: semi-ordered - affine geometries . The relation-theoretic calculus then also proves to be effective in more general geometric areas: affine line geometries , the main difference between which is that there are several connecting lines to two different points and affine lattices, i.e. affine line geometries with the existence of a maximum element in the The set of lines comprising a given line are synonymous with line relatives. Roland Soltysiak succeeded in demonstrating the synonymous connection between fast-affine relative 1st type and the almost- affine spaces induced by almost bodies , as well as with the almost- affine relative 2nd type to Sperner's spaces via almost bodies - i.e. , fully translatable affine geometries with distributive basis induced by regular almost vector spaces , along with the relational algebraic Description of the associated remote structures . ${\displaystyle {\mathfrak {B}}\in {\mathcal {R}}}$ ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$ ${\displaystyle {\mathfrak {A}}{\mathfrak {B}}}$ ${\displaystyle ({\mathcal {P}};\cdot ,{\mathfrak {E}})}$ ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$ ${\displaystyle ({\mathcal {P}};\cdot ,{\overline {.}},{\mathfrak {E}})}$

Operator operandum

The distance space according to operator operandum is expressed in relational language as follows: Starting from an affine relative , a projective multigroup is defined according to ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$ ${\displaystyle ({\mathcal {R}};\cdot ,{\mathfrak {E}})}$

${\displaystyle \left\{{\begin{array}{c}{\mathcal {R}}\times {\mathcal {R}}\longrightarrow \mathrm {Pot} \;{\mathcal {R}}\setminus \{\emptyset \}\\{\mathfrak {A,B}}\longmapsto {\mathfrak {A}}{\mathfrak {B}}:=\lbrace {\mathfrak {C}}\in {\mathcal {R}}\;\vert \;{\mathfrak {A}}\circ {\mathfrak {B}}\supset {\mathfrak {C}}\rbrace \end{array}}\right.}$

which in turn supplies the projective geometry of the distant space of - the functor is the geometry process - with the setting ${\displaystyle (\mathbb {P} ,\mathbb {G} ,\in )}$ ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})\gamma }$ ${\displaystyle \;\gamma }$

${\displaystyle \mathbb {P} :={\mathcal {R}}\setminus \{{\mathfrak {E}}\}\;,\;\mathbb {G} :=\{{\mathfrak {A}}{\mathfrak {B}}\cup \{{\mathfrak {A}},{\mathfrak {B}}\}\,\vert \,{\mathfrak {A}}\neq {\mathfrak {B}}\in \mathbb {P} \}}$

Constructive expandability

The described consonance of geometrical and (relational) algebraic speech can then be "constructively expanded" for important geometrical inference sentences by suitable calculation rules: The two-stage affine (H2) homogeneity rule proves to be an important rule,

${\displaystyle (H_{2})\bigwedge _{A,B,C\in {\mathfrak {P}}}:AB\subset AC\circ CB.}$

on the geometrical side it is equivalent to the Tamaschke axiom , which ensures the compatibility of the parallelism with the intersection point. The associative law valid in projective multigroups is equivalent to the axiom of homogeneity named after Oswald Veblen and John Wesley Young , which ensures the intersection of two straight lines lying in a projective plane. If three-digit affine relatives are used as a basis, these also turn out to be synonymous with the affine geometries, see the transition method mentioned above.

With the application of a projection functor to positions 1 and 4 in the concatenation product of two three-digit relations ${\displaystyle {\mathcal {F}}_{1,4}}$ ${\displaystyle \boxminus }$

${\displaystyle AXY\oslash XYB:=(AXY\boxminus XYB){\mathcal {F}}_{1,4}:=}$

${\displaystyle \{(C_{1},C_{2},C_{3},C_{4})\mid (C_{1},C_{2},C_{3})\in AXY\;\wedge \;(C_{2},C_{3},C_{4})\in XYB\}{\mathcal {F}}_{1,4}}$

the three-stage affine (H3) -homogeneity rule applies

${\displaystyle (H_{3})\;\bigwedge _{A,B,X,Y\in {\mathfrak {P}}}:AB\subset AXY\oslash XYB}$

if and only if, geometrically, the large affine theorem of Desargues is valid as an additional axiom.

The primary operation of the now three-digit relations provides parallel perspectives, for the secondary operation according to the “operator operandum” a so-called “inner relation product” is explained which generalizes the relation product . This product again provides two-digit relations that lie in a binary affine relative as a projection of the original three-digit relative. In the distance space, the calculation of the inner relational product corresponds to the construction of the 6th Hessenberg or Veblenian (distance) point given by five, which belong from the two-digit projections of the three-digit factors of an inner product. ${\displaystyle \oslash }$

Despite the synonymous relationship to the projective planes, the projective multigroups are not yet suitable as a calculus for intersection theorems such as Desargues' theorem, only the 2x2 relations operating on the Cartesian product of the basic set and the projective relatives derived therefrom according to the following transition provide a remedy : ${\displaystyle ({\mathcal {P}}_{2},{\mathcal {R}}_{2})}$

${\displaystyle ({\mathcal {P}};\cdot ,{\mathfrak {E}})}$be a projective multigroup with a base of 3 points, i.e. H. three elements . ${\displaystyle \mathbb {P} :={\mathcal {P}}\setminus \{{\mathfrak {E}}\}}$

The (2x2) -relations are called binary relations on the set according to and over ${\displaystyle {\mathcal {P}}_{2}:=\left\{{\binom {\mathfrak {C^{1}}}{\mathfrak {C^{2}}}}\;\vert \;{\mathfrak {C^{i}}}\in {\mathcal {P}},i=1,2\right\}}$ ${\displaystyle {\mathcal {R}}_{2}:=\left\{{\frac {\mathfrak {A}}{\mathfrak {B}}}\;\vert \;{\mathfrak {A}},{\mathfrak {B}}\in {\mathcal {P}}\right\}}$ ${\displaystyle {\binom {\mathfrak {C_{1}^{1}}}{\mathfrak {C_{1}^{2}}}}{\frac {\mathfrak {A}}{\mathfrak {B}}}{\binom {\mathfrak {C_{2}^{1}}}{\mathfrak {C_{2}^{2}}}}}$

${\displaystyle {\mathfrak {C_{1}^{i}}}{\mathfrak {A}}\ni {\mathfrak {C_{2}^{i}}};\;i=1,2}$

${\displaystyle {\mathfrak {C_{i}^{1}}}{\mathfrak {B}}\ni {\mathfrak {C_{i}^{2}}};\;i=1,2}$

Are defined.

The projective (H2x2) homogeneity rule

${\displaystyle (H_{2x2})\;\bigwedge _{{\mathfrak {A_{i}}},{\mathfrak {B}}\in {\mathcal {P}}}:{\mathfrak {A_{1}}}\in {\mathfrak {A_{2}}}{\mathfrak {A_{3}}}\;\;\Rightarrow \;\;{\frac {\mathfrak {A_{1}}}{\mathfrak {B}}}\;\subset \;{\frac {\mathfrak {A_{2}}}{\mathfrak {B}}}\;\circ \;{\frac {\mathfrak {A_{3}}}{\mathfrak {B}}}}$

then reversibly unambiguously corresponds to the validity of Desargues' large theorem in a projective plane and a modification of (H2x2) corresponds to Hessenberg's small invariance theorem . ${\displaystyle ({\mathcal {P}};\cdot ,{\mathfrak {E}})\gamma }$

Action theory

With relation-theoretic groupings, Arnold succeeds in a mathematical description of the concept of the action scheme, which was further developed by Jean Piaget and Hans Aebli , and which is represented on the background of the predicative expression for actions through forms of statement. Binary fields of action based on this are triples consisting of a non-empty set of objects , a non-empty set of parameters and a linguistic structure which, by inserting any parameter, becomes a verb through which ordered tuples of objects are related. In a binary field of action there is a two-digit form of statement for each , which is useful when inserted into the spaces , i.e. H. that the result of such an appointment can in principle be checked for truth. An example of this would be directions as parameter elements, read “comes in the direction ”, the statement form was “one comes in the direction from to ”. The fields of action defined in this way then turn out to be synonymous with binary relatives, here with action relatives when it is set: with . ${\displaystyle ({\mathfrak {P}},v(\cdot ),{\mathcal {P}})}$ ${\displaystyle \textstyle {\mathfrak {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle v(\cdot )}$ ${\displaystyle p\in {\mathcal {P}}}$ ${\displaystyle v(p)}$ ${\displaystyle A,C\in {\mathfrak {P}}}$ ${\displaystyle ({\mathfrak {P}},v(\cdot ),{\mathcal {P}})}$ ${\displaystyle p\in {\mathcal {P}}}$ ${\displaystyle \textstyle {\text{Man}}\;v(p)x_{0}x_{1}}$ ${\displaystyle A_{0},A_{1}\in {\mathfrak {P}}}$ ${\displaystyle x_{0},x_{1}}$ ${\displaystyle (\ast )}$ ${\displaystyle v(\cdot )}$ ${\displaystyle (\cdot )\,}$ ${\displaystyle (*)}$ ${\displaystyle p}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{1}}$ ${\displaystyle ({\mathfrak {P}},v(\cdot ),{\mathcal {P}})}$ ${\displaystyle ({\mathfrak {P}},{\mathcal {R}})}$ ${\displaystyle \textstyle {\mathcal {R}}:=\{{\dot {p}}\,|\,p\in {\mathcal {P}}\}}$ ${\displaystyle A_{0}{\dot {p}}A_{1}\;:\Leftrightarrow \,{\text{Man}}\;v(p)A_{0}A_{1}\;{\text{f.a.}}\;A_{0},A_{1}\in {\mathfrak {P}},p\in {\mathcal {P}}}$

Systems theory

The procedure chosen for affine geometries of assigning a relation of an affine relative to a parallel set is applied analogously to the construction of a system-describing relative: the fixed values ​​of a given system of engineering control theory become binary relations on the Cartesian product of the time and state set of the system assigned whose product leads to the control functions at suitable time intervals. The rule-relatives thus defined by Arnold allow him a synonymous identification of the abstract concept of system by Eduardo Sontag, which is based on the definition of a dynamic system by his teacher Rudolf Kálmán .

Synonymous context

One speaks of a general dynamic system according to Sontag and Kalman, if the following are given: ${\displaystyle ({\mathcal {T}},X,{\mathcal {U}},\Phi )}$

• as "amount of time" a subgroup  ; ${\displaystyle {\mathcal {T}}(+)<\mathbb {R} (+)}$
• a non-empty set whose elements are called “states”; ${\displaystyle X}$
• a non-empty set , the elements of which are called "control values"; ${\displaystyle {\mathcal {U}}}$
• a state transfer function ${\displaystyle \Phi :{\mathcal {D}}_{\Phi }\longrightarrow X,}$

where is a subset of ${\displaystyle {\mathcal {D}}_{\Phi }}$ ${\displaystyle \{(\tau ,\sigma ,x,\omega )\mid \sigma ,\tau \in {\mathcal {T}};\sigma \leq \tau ;x\in X;\omega \in {\mathcal {U}}^{[\sigma ,\tau )}\}}$

and if the following conditions are met:

1. Non-triviality: For every state there is at least one pair in and one such that “ is applicable to”, i. H. such that ${\displaystyle x_{0}\in X}$ ${\displaystyle \sigma <\tau }$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle \omega \in {\mathcal {U}}^{[\sigma ,\tau )}}$ ${\displaystyle \omega }$ ${\displaystyle x_{0}}$ ${\displaystyle (\tau ,\sigma ,x_{0},\omega )\in {\mathcal {D}}_{\Phi }}$
2. Restriction: If applicable to , then the restriction to is also applicable to and the restriction is applicable to . ${\displaystyle \omega \in {\mathcal {U}}^{[\sigma ,\tau )}}$ ${\displaystyle x}$ ${\displaystyle \mu \in [\sigma ,\tau )}$ ${\displaystyle \omega _{1}=\omega \mid _{[\sigma ,\mu )}\in {\mathcal {U}}^{[\sigma ,\mu )}}$ ${\displaystyle x}$ ${\displaystyle \omega _{2}=\omega \mid _{[\mu ,\tau )}\in U^{[\mu ,\tau )}}$ ${\displaystyle x(\mu )=(\mu ,\sigma ,x,\omega _{1})\Phi }$
3. Semigroup: If there are three real numbers with , is and and is a state with , then the concatenation is applicable to, and it holds ${\displaystyle \sigma ,\mu ,\tau }$ ${\displaystyle \sigma <\mu <\tau }$ ${\displaystyle \omega _{1}\in {\mathcal {U}}^{[\sigma ,\mu )}}$ ${\displaystyle \omega _{2}\in {\mathcal {U}}^{[\mu ,\tau )}}$ ${\displaystyle x}$ ${\displaystyle (\mu ,\sigma ,x,\omega _{1})\Phi =x_{1}\wedge (\tau ,\mu ,x_{1},\omega _{2})\Phi =x_{2}}$ ${\displaystyle \omega =\omega _{1}{\dot {\vee }}\omega _{2}\in {\mathcal {U}}^{[\sigma ,\tau )}}$ ${\displaystyle x}$ ${\displaystyle (\tau ,\sigma ,x,\omega )\Phi =x_{2}.}$
4. Identity: For each and every one , the blank figure on is applicable, and it holds ${\displaystyle \sigma \in {\mathcal {T}}}$ ${\displaystyle x\in X}$ ${\displaystyle \diamond \in {\mathcal {U}}^{[\sigma ,\sigma )}}$ ${\displaystyle x}$ ${\displaystyle (\sigma ,\sigma ,x,\diamond )\Phi =x.}$
5. Reduction: Let it be and it applies , where it is set, it follows . ${\displaystyle u_{1},u_{2}\in {\mathcal {U}}}$ ${\displaystyle \bigwedge _{\sigma \in {\mathcal {T}}}\bigwedge _{\tau \in {\mathcal {T}}}\bigwedge _{x\in X}(\tau ,\sigma ,x,\omega _{u_{1}})\Phi =(\tau ,\sigma ,x,\omega _{u_{2}})\Phi }$ ${\displaystyle \omega _{u_{i}}(t)\equiv u_{i}\;{\text{ f.a. }}t\in [\sigma ,\tau )}$ ${\displaystyle u_{1}=u_{2}}$

Example:

A linear time-invariant multivariable system that is described by

${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$

with as well as matrices and with correspondingly adapted dimensions and initial value , creates a relationship between the states , their temporal derivatives and inputs at every point in time . With the help of the matrix exponential function for the solution of the vector valued differential equation, a Sontag system is defined as follows: ${\displaystyle {\dot {x}}(t)\in \mathbb {R} ^{n},u(t)\in \mathbb {R} ^{m}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle x(t_{0})=x_{0}}$ ${\displaystyle t}$ ${\displaystyle x(t)}$ ${\displaystyle {\dot {x}}(t)}$ ${\displaystyle u(t)}$

${\displaystyle {\begin{cases}{\mathcal {T}}(+)\ :=\ \mathbb {R} (+)\\X\ :=\ \mathbb {R} ^{n}\\{\mathcal {U}}\ :=\ \mathbb {R} ^{m}\\{\mathcal {D}}_{\Phi }\ :=\ \{(\tau ,\tau _{0},x_{0},u(\cdot ))\mid \tau _{0}\leq \tau ,\ u(\cdot )\ {\text{stückweise stetig differenzierbar}}\}\\(\tau ,\tau _{0},x_{0},u(\cdot ))\Phi \ :=\ e^{A(\tau -\tau _{0})}x_{0}\,+\,\int _{\tau _{0}}^{\tau }e^{A(\tau -\sigma )}Bu(\sigma )\,d\sigma \end{cases}}}$

One speaks of a rule-relative when the following are given: ${\displaystyle ({\mathfrak {P}},{\mathcal {R}},\Pi \cdot dt),}$

• as "amount of time" a subgroup ${\displaystyle {\mathcal {T}}(+)<\mathbb {R} (+);}$
• a non-empty set whose elements are called “states”; ${\displaystyle X}$
• a non-empty set of binary relations on the basic set ${\displaystyle {\mathcal {R}}}$ ${\displaystyle {\mathfrak {P}}={\mathcal {T}}\times X:{\mathcal {R}}\subset \mathrm {Pot} ({\mathfrak {P}}\times {\mathfrak {P}});}$
• a group of figures , in which each pair has a figure ${\displaystyle \Pi \cdot dt}$ ${\displaystyle \sigma ,\tau \in {\mathcal {T}}}$ ${\displaystyle \sigma \leq \tau }$

${\displaystyle \Pi _{\sigma }^{\tau }\cdot dt\colon \left\{{\begin{array}{c}{\mathcal {R}}^{[\sigma ,\tau )}\longrightarrow \mathrm {Pot} ({\mathfrak {P}}\times {\mathfrak {P}})\\\Omega \longmapsto \Pi _{\sigma }^{\tau }\Omega dt\end{array}}\right.}$

exists

and if the following properties apply:

1. ${\displaystyle \bigwedge _{x\in X}\bigvee _{\sigma <\tau }\bigvee _{\Omega \in {\mathcal {R}}^{[\sigma ,\tau )}}(\sigma ,x)\Pi _{\sigma }^{\tau }\Omega dt\neq \emptyset }$. A relation can be applied to every point or state.
2. ${\displaystyle (\sigma ,x)\Pi _{\sigma }^{\tau }\Omega dt(\tau ,y_{1})\wedge (\sigma ,x)\Pi _{\sigma }^{\tau }\Omega dt(\tau ,y_{2})\succ y_{1}=y_{2}}$. Clearly means that, given the same initial values, manipulated variable progressions and time segments, the state reached is unambiguous.
3. ${\displaystyle \Pi _{\sigma }^{\mu }\Omega _{1}dt\circ \Pi _{\mu }^{\tau }\Omega _{2}dt=\Pi _{\sigma }^{\tau }(\Omega _{1}{\dot {\vee }}\Omega _{2})dt}$. Describes how a manipulated variable curve can be composed of two individual curves.
4. ${\displaystyle \bigwedge _{c\in {\mathcal {R}}}\Pi _{\sigma }^{\tau }cdt=c\cap (\tau -\sigma ),}$where the relation explained according to is and is to be set for fa . Ensures the consistency of the relations to the set of relations generated from constant control functions . ${\displaystyle (\tau -\sigma )}$ ${\displaystyle (\sigma ',x)(\tau -\sigma )(\tau ',y)\;:\Leftrightarrow \;\tau '=\tau \wedge \sigma '=\sigma }$ ${\displaystyle \Pi _{\sigma }^{\tau }c\,dt=\Pi _{\sigma }^{\tau }\Omega _{c}dt}$ ${\displaystyle \Omega _{c}(t)\equiv c\;}$ ${\displaystyle t\in [\sigma ,\tau )}$ ${\displaystyle [\Pi \cdot dt]}$
5. ${\displaystyle \bigwedge _{\sigma \in {\mathcal {T}}}\bigwedge _{x\in X}(\sigma ,x)\Pi _{\sigma }^{\sigma }\diamond dt(\sigma ,x)}$applies to the empty figure . Is only of a formal nature for the transfer of dynamic systems to relative ones. ${\displaystyle \diamond \in {\mathcal {R}}^{[\sigma ,\sigma )}}$

The linear time-invariant system mentioned above generates a rule relative as follows:

1. Determination of the basic amount , (at ). ${\displaystyle {\mathfrak {P}}:=\mathbb {R} \times \mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {T}}=\mathbb {R} }$
2. Definition of the system described control relations and for constant control values . ${\displaystyle {\mathfrak {C}}\in {\mathcal {R}}}$ ${\displaystyle (t_{0},x_{0})\langle u\rangle (t_{1},x_{1})\,:\Leftrightarrow \,e^{A(t_{1}-t_{0})}x_{0}\,+\,\int _{t_{0}}^{t_{1}}e^{A(t-\tau )}Bu\,d\tau =x_{1}\,\Leftrightarrow \,\bigvee _{x(t)}{\dot {x}}(t)=Ax(t)+Bu(t)\quad \wedge \quad x(t_{0})=x_{0}\quad \wedge \quad x(t_{1})=x_{1}}$
3. Piecewise constant control functions are then given with the help of the simple relation product according to Axiom 3, for arbitrarily variable functions this happens through the border crossing in the map . ${\displaystyle \circ }$ ${\displaystyle [\Pi \cdot dt]}$

For all rule relatives and all systems there are suitable transition procedures and for a synonymous connection: ${\displaystyle ({\mathfrak {P}}={\mathcal {T}}\times X,{\mathcal {R}},\Pi \cdot dt)}$ ${\displaystyle ({\mathcal {T}},X,{\mathcal {U}},\Phi )}$ ${\displaystyle \Gamma }$ ${\displaystyle \Sigma }$

${\displaystyle ({\mathfrak {P}}={\mathcal {T}}\times X,{\mathcal {R}},\Pi \cdot dt)\Sigma \Gamma =({\mathfrak {P}}={\mathcal {T}}\times X,{\mathcal {R}},\Pi \cdot dt).}$

${\displaystyle ({\mathcal {T}},X,{\mathcal {U}},\Phi )\Gamma \Sigma \cong ({\mathcal {T}},X,{\mathcal {U}},\Phi ).}$

Further results and constructive expandability

The system-theoretical properties of controllability / accessibility, observability / distinguishability and zero dynamics, which are important for dealing with control-related questions , are fully formulated by output-rule-relatives by Marc Schleuter and Markus Lemmen. In fuzzy logic -based schemes of control systems with fuzzy state relative to the fuzzy control described by the supporting points quantity and with Fuzyy-time state relative to the fuzzy control after the time component, including its coupling than double relative. The methodology of rule-relatives for continuous systems also provides synonymous descriptions for linear discrete-time systems with linear rule-relatives, are these time-invariant and commutative, the associated geometry is weakly affine and satisfies Desargues' little theorem. In a relational algebraic consideration of special classes of state-homogeneous and input-homogeneous bilinear systems , Axel Sauerland demonstrated that the differential equation relatives describing the solution space are affine relatives and even satisfy the large affine theorem of Desargues. Arranged affine geometries can in turn be generated as a special case of rule relatives restricted by suitable additional axioms.

literature

• H.-J. Arnold , W. Benz , H. Wefelscheid (eds.): Contributions to geometric algebra. In: Proceedings of the Symposium on Geometric Algebra from March 29 to April 3, 1976 in Duisburg. Birkhäuser, Basel 1977, ISBN 3-0348-5573-7 . doi: 10.1007 / 978-3-0348-5573-0 .
• H.-J. Arnold, W. Junkers , W. Kühnel , G. Törner , H. Wefelscheid (eds.): Contributions to geometric algebra and its applications. In: Proceedings of the 2nd Duisburg Symposium on Geometric Algebra and its Applications. University of Duisburg, 1987.
• E. Heineken et al .: Strategies of thinking in the regulation of a simple dynamic system under different dead time conditions. In: Language & Cognition. 11/1986, pp. 136-148.
• D. Hilbert: Fundamentals of Geometry. 13th edition. Teubner, Stuttgart 1987, ISBN 3-519-00237-X . (Copy of the 1903 edition, first edition 1899, archive.org ).
• RC Lyndon : Relation algebras and projective geometries. In: Michigan Math. J. 8, 1961, pp. 21-28, doi: 10.1307 / mmj / 1028998510
• SE Schmidt: Fundamentals of a general affine geometry. Springer Verlag, 1995. doi: 10.1007 / 978-3-0348-9233-9

Individual evidence

1. ^ H. Grassmann: About the place of the Hamiltonian quaternions in the expansion theory. In: Mathematische Annalen, 1877.
2. ^ WK Clifford: On the classification of geometric algebras. In: R. Tucker (Ed.): Mathematical Papers. Macmillian, London 1882, pp. 397-401.
3. ^ E. Artin: Geometric Algebra . Interscience Publishers. New York 1957.
4. ^ M. Hall: Projective Planes. In: Trans. Amer. Math. Soc. 54, 1943, pp. 229-277.
5. ^ E. Sperner: Affine Spaces with Weak Incidence and Associated Algebraic Structures. In: Journal for pure and applied mathematics. 204, 1960, pp. 205-215. doi: 10.1515 / crll.1960.204.205 .
6. ↑ W. Prenowitz: Projective geometries as multi groups. In: Amer. J. Math. (65) 1943, pp. 235-256.
7. ↑ H.-J. Arnold: The projective closure of affine geometries with the help of relation-theoretic methods. In: Treatises from the Mathematical Seminar of the University of Hamburg. University of Berlin, Hamburg (40) 1974, pp. 197-214. doi: 10.1007 / BF02993598 .
8. ↑ H.-J. Arnold: Directional algebras. In: Contributions to Geometry. 1979, pp. 379-382. doi : 10.1007 / 978-3-0348-5765-9_22 .
9. ↑ H.-J. Arnold: About the assumption of the didactic relevance of a new geometrical-algebraic axiomatic of arrangement. In: Mitteilungen der mathem. Society in Hamburg. Volume X Issue 6, 1978.
10. ↑ T. Wey: Non-linear control systems. A differential algebraic approach. BG Teubner, Stuttgart 2002.
11. ↑ H. Schwarz: Nonlinear Control Systems - System Theory Basics. Oldenburg, Munich 1991.
12. ↑ H.-J. Arnold: The system concept of control theory and rule-relatives. In: Results in Mathematics. Birkhäuser, Basel (28) 1995, pp. 195-208. doi: 10.1007 / BF03322252 .
13. ^ HJ Zimmermann: Fuzzy Sets, Decision Making and Expert Systems. Kluwer, Boston / USA 1987. doi: 10.1007 / 978-94-009-3249-4 .
14. ↑ H.-J. Arnold: A relation-theoretic algebraization of arranged affine and projective geometries. In: Treatises from the Mathematical Seminar of the University of Hamburg. Universität Berlin, Hamburg (45), 1976, pp. 3-60. doi: 10.1007 / BF02992902 .
15. ↑ T. Ledabo: Relational characterization of partially arranged weak-affine geometries and their remote structures . Dissertation 1997. University of Duisburg, DNB 976105306 .
16. ↑ CM Senevirathne: Projection of semi-ordered affine geometries using relational algebraic methods. In: Treatises from the Mathematical Seminar of the University of Hamburg. Universität Berlin, Hamburg (62) 1992, pp. 65-80. doi: 10.1007 / BF02941619 .
17. ↑ H. Karzel, H. Lenz: About Hilbertsche and Spernersche arrangement. In: Treatises from the Mathematical Seminar of the University of Hamburg. Universität Berlin, Hamburg, (25) 1961, pp. 82-88. doi: 10.1007 / BF02992778 .
18. ↑ A. Kopp: Development of relation-theoretical tools for algebraization and construction of general affine structures . Dissertation 1986. University of Duisburg.
19. ^ J. Andre: Affine geometries over almost bodies. In: With. Math. Sem. Giessen. (114) 1975, pp. 1-99.
20. ↑ H.-J. Arnold: Algebraic and geometric characterization of the weakly affine vector spaces over almost fields. In: Treatises from the Mathematical Seminar of the University of Hamburg. University of Berlin, Hamburg (32) 1968, pp. 73-88. doi: 10.1007 / BF02993915 .
21. ↑ R. Soltysiak, H.-J. Arnold: The projection of affine structures over almost bodies with the help of relation-theoretic methods. In: Results in Mathematics. Volume 4. Birkhäuser, Basel 1981, pp. 119–121. doi: 10.1007 / BF03322971
22. ↑ H.-J. Arnold: Affine Relatives. In: Results in Mathematics. Birkhäuser, Basel (12) 1987, pp. 1–26. doi: 10.1007 / BF03322375
23. ↑ H.-J. Arnold: About a relational calculus for the algebraization of projective levels. In: Results in Mathematics. Birkhäuser, Basel (19) 1991, pp. 211-236. doi: 10.1007 / BF03323282
24. ↑ H.-J. Arnold: Comments on Hessenberg's small invariance theorem. (= Series of publications by the Department of Mathematics / Gerhard Mercator University Duisburg University. 372). 1997, DNB 95009191X .
25. ↑ H.-J. Arnold: Relational groupings in the context of Piaget's developmental psychology. In: Contributions to Geometric Algebra. Birkhäuser, Basel 1977, pp. 361–366. doi : 10.1007 / 978-3-0348-5573-0_49 .
26. ↑ H. Aebli: Thinking, the ordering of doing . Volume 1: Cognitive Processes of Action Theory. Stuttgart: Klett-Cotta 1980.
27. ^ J. Piaget: Recherches sur l'abstraction reflechissante . 1. L'abstraction des relations logico-arithmetiques, Presses universitaires de France, Paris (1977).
28. ↑ H.-J. Arnold: On the genesis of math in suitable fields of action . (= Series of publications by the Department of Mathematics / Gerhard Mercator University Duisburg University. 196). 1991.
29. ↑ H.-J. Arnold: Geometric relation algebra and Piaget's theorem of patterns (schemes) . (= Series of publications by the Department of Mathematics / Gerhard Mercator University Duisburg University. 423) 1998.
30. ↑ H.-J. Arnold: On the idea of ​​mathematising in PIAGET's theory of cognitive processes . (= Series of publications by the Department of Mathematics / Gerhard Mercator University Duisburg University. 449). 1999.
31. ↑ H.-J. Arnold: For the mathematical description of goal-oriented human actions on technical systems . (= Series of publications by the Department of Mathematics / Gerhard Mercator University Duisburg University. 173). 1990.
32. ^ ED Sontag: Mathematical Control Theory. Deterministic Finite Dimensional Systems. 2nd Edition. Springer, Berlin 1998.
33. ^ RE Kalman, PL Falb, MA Arbib: Topics in mathematical system theory . New York 1969.
34. ↑ M. Lemmen, M. Schleuter: Relational Control Structures. In: Results in Mathematics. Birkhäuser, Basel (29) 1996, pp. 100-110. doi: 10.1007 / BF03322209 .
35. ↑ H.-J. Arnold: The double relative of a control system . (= Series of publications by the Department of Mathematics / Gerhard Mercator University Duisburg University. 350). 1996.
36. ↑ D. Wetscheck: Fuzzification of control systems by means of relational algebraic and graph theoretical methods . Dissertation 1999. University - Comprehensive University Duisburg.
37. ↑ P. Stemper: Relational construction of weakly affine geometries from linear control systems . Dissertation 1997. University - Comprehensive University Duisburg.
38. ↑ A. Sauerland: Relative differential equations of classes of linear and non-linear control systems. Dissertation 1994. University - Comprehensive University of Duisburg.