Pseudo magma

This article deals with the mathematical structure pseudo-magma, further meanings under pseudo-magma (disambiguation) .

A pseudo-magma (neuter , plural pseudo magmas ), partial Magma , pseudo-groupoid , partial groupoid or Halbgruppoid (based on the English halfgroupoid ) is an algebraic structure (more precisely, partial algebra ) consisting of a quantity of one, and partial mapping consists. ${\displaystyle (M,f)}$ ${\displaystyle M}$ ${\displaystyle f\colon M\times M\rightsquigarrow M}$

It is a generalization of the mathematical concept of the magma or groupoid , in which the mapping must be a two-digit, internal link ( ), i.e. it must no longer be partial. ${\displaystyle f}$ ${\displaystyle f\colon M\times M\rightarrow M}$

Alternative definition

A pseudo-magma can also be defined as a set together with an outer two-digit link of the second kind . ${\displaystyle (M,f)}$ ${\displaystyle M}$ ${\displaystyle f\colon M\times M\rightarrow B}$

A pseudo-magma defined via a partial mapping can be converted into an equivalent pseudo-magma with an outer two-digit link of the second kind by specifying with and setting, if , otherwise . ${\displaystyle (M,f)}$ ${\displaystyle (M,f')}$ ${\displaystyle B:=M\cup \{x\}}$ ${\displaystyle x\notin M}$ ${\displaystyle f'(a,b)=x}$ ${\displaystyle (a,b)\notin {\mbox{Def}}(f)}$ ${\displaystyle f'(a,b)=f(a,b)}$

Conversely, a pseudo-magma with an outer two-digit link of the second kind can be converted into an equivalent pseudo-magma defined via a partial mapping by setting as undefined at the point , if , otherwise . ${\displaystyle (M,f')}$ ${\displaystyle (M,f)}$ ${\displaystyle f}$ ${\displaystyle (a,b)}$ ${\displaystyle f'(a,b)\notin M}$ ${\displaystyle f(a,b)=f'(a,b)}$

Both definitions are therefore equivalent in a certain sense.

Further definitions

Sub-pseudomagma

Analogous to a sub magma or a sub-group , a lower pseudo magma (or Teilhalbgruppoid or Unterhalbgruppoid based on the English subhalfgroupoid be defined) from a pseudo-magma. Here, however, the definition range of the link must be considered separately.

Be a pseudo-magma. A pseudo-magma is called a sub-pseudomagma of , if and , i.e. H. the shortcut is the restriction of on . ${\displaystyle {\boldsymbol {M}}=(M,*)}$ ${\displaystyle {\boldsymbol {U}}=(U,\circ )}$ ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle U\subseteq M}$ ${\displaystyle \circ =*|_{{\mbox{Def}}(\circ )}}$ ${\displaystyle \circ }$ ${\displaystyle *}$ ${\displaystyle {\mbox{Def}}(\circ )\subseteq U\times U}$

So if and is a sub-pseudo magma of when and it is ${\displaystyle {\boldsymbol {U}}}$ ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle U\subseteq M}$

${\displaystyle {\mbox{Def}}(\circ )\subseteq {\mbox{Def}}(*)\cap U\times U}$

and

${\displaystyle a*b=a\circ b\in U}$for everyone . ${\displaystyle (a,b)\in {\mbox{Def}}(\circ )}$

A magma can therefore contain a sub-pseudomagma that is not a sub- magma , namely if: ${\displaystyle {\boldsymbol {M}}=(M,*)}$

${\displaystyle {\mbox{Def}}(\circ )\subsetneq U\times U={\mbox{Def}}(*)\cap U\times U}$.

example

Be and with , and following shortcut boards for and : ${\displaystyle {\boldsymbol {M}}=(M,*)}$ ${\displaystyle {\boldsymbol {U}}=(U,\circ )}$ ${\displaystyle M=\{a,b,c\}}$ ${\displaystyle U=\{a,b\}}$ ${\displaystyle *}$ ${\displaystyle \circ }$

${\displaystyle *}$ a b c a a b - b c b a c c a -

${\displaystyle \circ }$ a b a a b b - b

Then sub-pseudomagma is of . ${\displaystyle {\boldsymbol {U}}}$ ${\displaystyle {\boldsymbol {M}}}$

Remarks:

• The value of can be anything (it could also be , or undefined), there . ${\displaystyle b*a}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle (b,a)\notin {\mbox{Def}}(\circ )}$
• However, if there were, then there would be no sub-pseudomagma of , since then because of would not apply. ${\displaystyle (a,b)\notin {\mbox{Def}}(*)}$ ${\displaystyle {\boldsymbol {U}}}$ ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle (a,b)\in {\mbox{Def}}(\circ )}$ ${\displaystyle {\mbox{Def}}(\circ )\subset {\mbox{Def}}(*)\cap U\times U}$

Further properties of sub-pseudomagmas

Be a pseudo-magma and a sub -pseudo-magma of . ${\displaystyle {\boldsymbol {M}}=(M,*)}$ ${\displaystyle {\boldsymbol {U}}=(U,\circ )}$ ${\displaystyle {\boldsymbol {M}}}$

• ${\displaystyle {\boldsymbol {U}}}$means closed in (English "closed in M" ), if from and and follows and . Example: ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle a,b\in U}$ ${\displaystyle (a,b)\in {\mbox{Def}}(*)}$ ${\displaystyle a*b=c}$ ${\displaystyle (a,b)\in {\mbox{Def}}(\circ )}$ ${\displaystyle a\circ b=c}$

${\displaystyle *}$ a b c a a b - b b - a c c a -

${\displaystyle \circ }$ a b a a b b b -

• ${\displaystyle {\boldsymbol {M}}}$ie extension of (Engl. "extension of U" ) when made and followed , and out of following . Example: ${\displaystyle {\boldsymbol {U}}}$ ${\displaystyle a,b\in M}$ ${\displaystyle (a,b)\in {\mbox{Def}}(*)}$ ${\displaystyle a,b\in U}$ ${\displaystyle c\in M\land c\notin U}$ ${\displaystyle \exists (a,b)\in U\times U:(a,b)\in {\mbox{Def}}(*)\land c=a*b}$

${\displaystyle *}$ a b c a a - - b b c - c - - -

${\displaystyle \circ }$ a b a a - b b -

• An extension of means full expansion of (Engl. "Complete extension of U" ) if . Example: ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle {\boldsymbol {U}}}$ ${\displaystyle {\boldsymbol {U}}}$ ${\displaystyle \forall (a,b)\in U\times U:(a,b)\in {\mbox{Def}}(*)}$

${\displaystyle *}$ a b c a a b - b b c - c - - -

${\displaystyle \circ }$ a b a a - b b -

• An extension of means open extension of (Engl. "Open extension of U" ) when made , and follows and , and from and follow . Example: ${\displaystyle {\boldsymbol {M}}}$ ${\displaystyle {\boldsymbol {U}}}$ ${\displaystyle {\boldsymbol {U}}}$ ${\displaystyle (a,b)\in {\mbox{Def}}(*)}$ ${\displaystyle a*b=c}$ ${\displaystyle c\in U}$ ${\displaystyle (a,b)\in {\mbox{Def}}(\circ )}$ ${\displaystyle a\circ b=c}$ ${\displaystyle c\in M\land c\notin U}$ ${\displaystyle (a,b)\in {\mbox{Def}}(*)\land (a',b')\in {\mbox{Def}}(*)\land a*b=a'*b'=c}$ ${\displaystyle a=a'\land b=b'}$

${\displaystyle *}$ a b c d a a - - - b d c - - c - - - - d - - - -

${\displaystyle \circ }$ a b a a - b - -

Remarks:

• Each pseudo-magma is a self-contained sub-pseudo-magma.
• Each pseudo-magma is an open extension of itself.
• A pseudo-magma that is a complete extension of itself is a magma.
• A non-magma pseudo-magma can have an open, or full, or open and full extension. Example of an open and full extension:

${\displaystyle *}$ a b c d e a a e - - - b d c - - - c - - - - - d - - - - - e - - - - -

${\displaystyle \circ }$ a b a a - b - -

Laws of Calculation

Analogous to a magma, a pseudo-magma can be associative or commutative , but here the domain of definition must be considered more precisely. The following modified calculation laws apply:

• A pseudo-Magma is associative (and is also called partial semigroup ) if for all with and applies ${\displaystyle (M,\circ )}$ ${\displaystyle x,y,z\in M}$ ${\displaystyle x\circ y\in M}$ ${\displaystyle y\circ z\in M}$

1. ${\displaystyle x\circ (y\circ z)\in M}$ exactly when ${\displaystyle (x\circ y)\circ z\in M}$
2. and , if (and thus after 1. also ) ${\displaystyle x\circ (y\circ z)=(x\circ y)\circ z}$ ${\displaystyle x\circ (y\circ z)\in M}$ ${\displaystyle (x\circ y)\circ z\in M}$

• A pseudo-magma is commutative if holds for all ${\displaystyle (M,\circ )}$ ${\displaystyle x,y\in M}$

1. ${\displaystyle x\circ y\in M}$ exactly when ${\displaystyle y\circ x\in M}$
2. and , if (and thus after 1. also ) ${\displaystyle x\circ y=y\circ x}$ ${\displaystyle x\circ y\in M}$ ${\displaystyle y\circ x\in M}$

Examples

• An example of an associative pseudo-magma can be found in the so-called small categories , in which the class of arrows is a lot. This set, together with the linkage explained for arrows, forms an associative pseudo-magma. - The formal requirement that the category has to be small is mostly negligible. As a rule, all knowledge about pseudo-magmas can also be transferred to the class of arrows with the associated link.
• In general, the requirements for categories can be reduced to a composition operation in order to obtain a pseudo-magma, but this then does not have to be associative and also has no single elements.
• Any set  M of images becomes an associative pseudo-magma by virtue of being executed one after the other as a composition ${\displaystyle \circ \colon M\times M\to M,(\operatorname {f} ,\operatorname {g} )\mapsto \operatorname {f} \circ \operatorname {g} }$ ${\displaystyle (M,\circ ).}$
• Formal languages are generally associative pseudo-magmas in terms of concatenation (spelling) as a link. The so-called * -language (read: "Star language", cf. Kleene star ) over an alphabet Σ is initially a semigroup (even a monoid ), because in it the concatenation of two words is explained to a new word and that new word is back in the language. Formal languages, however, are defined as any subsets of any such * -language, so that in a special language the concatenation of two words is still explained / explainable, but does not lead to a word in the same language. ${\displaystyle \Sigma ^{*}}$

Individual evidence

1. ^ ^{A b} Günther Eisenreich: Lexicon of Algebra . Akademie-Verlag / Springer-Verlag, Berlin 1989, ISBN 3-05-500231-8 .
2. ↑ ^{a b c d e f} Richard Hubert Bruck: A survey of binary systems . In: PJHilton (ed.): Results of mathematics and their border areas . 3. Edition. tape 20 . Springer Verlag, Berlin / Heidelberg / New York 1971, ISBN 978-3-662-42837-5 .
3. ^ Yoshifumi Inui, François Le Gall: Quantum Property Testing of Group Solvability . S. 2 , arxiv : 0712.3829 (definition at the beginning of § 2.1).
4. ↑ Shelp, RH: A Partial Semigroup Approach to Partially Ordered Sets . In: Proc. London Math. Soc. (1972) p3-24 (1) . London Mathematical Soc., 1972, pp. 46-58.