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 f \ colon M \ times M \ rightsquigarrow M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec777b2a84dfc8c2c266adcc29d9cadbcd09634)
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.
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![{\ displaystyle f \ colon M \ times M \ rightarrow M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58a6164ca51e8cadaef170a363928bf0b697d406)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fabc61051678478033383a59d8791e38158c455)
![{\ displaystyle f \ colon M \ times M \ rightarrow B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e411cab969c6076bac12b869c6329c0730c9bb1f)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fabc61051678478033383a59d8791e38158c455)
![{\ displaystyle (M, f ')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba856d34a6ffc969577a9bd63643c6ed000ca83)
![{\ displaystyle B: = M \ cup \ {x \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15044df2784e33ac095a6753f9ddd8c08c71405b)
![{\ displaystyle x \ notin M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f394c6b965971d6a1353c7a05bdeee74485a11ae)
![{\ displaystyle f '(a, b) = x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3056917b9fe2aacd94f1fdceaff4e99726912ef9)
![{\ displaystyle (a, b) \ notin {\ mbox {Def}} (f)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57df5f528794ca15d332c0204a641f25f4970fa1)
![{\ displaystyle f '(a, b) = f (a, b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5295b35cf34acca42af3cd1b5fa42280e5d7afbc)
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 ')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba856d34a6ffc969577a9bd63643c6ed000ca83)
![{\ displaystyle (M, f)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fabc61051678478033383a59d8791e38158c455)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![(from)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c)
![{\ displaystyle f '(a, b) \ notin M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79097df70919d625f1110d301a2dae7cb5c159fa)
![{\ displaystyle f (a, b) = f '(a, b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a54a4c3bec65609cb00b7be366d7196969955698)
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, *)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4adcaacc33b8bdd03038fffc913d00700e64ed65)
![{\ displaystyle {\ boldsymbol {U}} = (U, \ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04fc2ac102bf0e2677da14bdb8266ff10108b912)
![{\ displaystyle {\ boldsymbol {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d5057408f7628386ba91f52a93e6a31c0416)
![U \ subseteq M](https://wikimedia.org/api/rest_v1/media/math/render/svg/a57ecb017c8458e5f519dbca5ddb7fffde7f1a16)
![{\ displaystyle \ circ = * | _ {{\ mbox {Def}} (\ circ)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2484b4150481117fdd60a03560bfdd583046ba46)
![\ circ](https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31)
![*](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755)
![{\ displaystyle {\ mbox {Def}} (\ circ) \ subseteq U \ times U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae88e2b8d8b904cd39fd667909493e6661f925bd)
So if and is a sub-pseudo magma of when and it is
![{\ boldsymbol U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7400a9c60ebb8eda9fff860d7d8e1435fea8308d)
![{\ displaystyle {\ boldsymbol {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d5057408f7628386ba91f52a93e6a31c0416)
![U \ subseteq M](https://wikimedia.org/api/rest_v1/media/math/render/svg/a57ecb017c8458e5f519dbca5ddb7fffde7f1a16)
![{\ displaystyle {\ mbox {Def}} (\ circ) \ subseteq {\ mbox {Def}} (*) \ cap U \ times U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dce08ca55a65e82b0fad227e5d405af0b8f6340)
and
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for everyone .![{\ displaystyle (a, b) \ in {\ mbox {Def}} (\ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbdcbf86e3879bf82ba1aa4c212d55a7a9cecd5)
A magma can therefore contain a sub-pseudomagma that is not a sub- magma , namely if:
![{\ displaystyle {\ boldsymbol {M}} = (M, *)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4adcaacc33b8bdd03038fffc913d00700e64ed65)
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.
example
Be and with , and following shortcut boards for and :
![{\ displaystyle {\ boldsymbol {M}} = (M, *)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4adcaacc33b8bdd03038fffc913d00700e64ed65)
![{\ displaystyle {\ boldsymbol {U}} = (U, \ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04fc2ac102bf0e2677da14bdb8266ff10108b912)
![{\ displaystyle M = \ {a, b, c \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ffc603d371d94582949070298a13330a8405f8)
![{\ displaystyle U = \ {a, b \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e53f20f1b018553c06ef07da0a08b7d47fb1ce)
![*](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755)
![\ circ](https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31)
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Then sub-pseudomagma is of .
![{\ displaystyle {\ boldsymbol {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d5057408f7628386ba91f52a93e6a31c0416)
Remarks:
- The value of can be anything (it could also be , or undefined), there .
![{\ displaystyle b * a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55b63471d3f6565a671e765b2e6c22d02c9ca004)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![{\ displaystyle (b, a) \ notin {\ mbox {Def}} (\ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f38618ed0dcfc5e5a195356356f434dcab9a5d80)
- 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}} (*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05422c3d964f771a2ab15e5da8fafaf000eccf1d)
![{\ boldsymbol U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7400a9c60ebb8eda9fff860d7d8e1435fea8308d)
![{\ displaystyle {\ boldsymbol {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d5057408f7628386ba91f52a93e6a31c0416)
![{\ displaystyle (a, b) \ in {\ mbox {Def}} (\ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbdcbf86e3879bf82ba1aa4c212d55a7a9cecd5)
![{\ displaystyle {\ mbox {Def}} (\ circ) \ subset {\ mbox {Def}} (*) \ cap U \ times U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ece0ae7791bec4907734f09abdedf1a971e28c5)
Further properties of sub-pseudomagmas
Be a pseudo-magma and a sub -pseudo-magma of .
![{\ displaystyle {\ boldsymbol {M}} = (M, *)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4adcaacc33b8bdd03038fffc913d00700e64ed65)
![{\ displaystyle {\ boldsymbol {U}} = (U, \ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04fc2ac102bf0e2677da14bdb8266ff10108b912)
![{\ displaystyle {\ boldsymbol {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d5057408f7628386ba91f52a93e6a31c0416)
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means closed in (English "closed in M" ), if from and and follows and . Example:![{\ displaystyle {\ boldsymbol {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d5057408f7628386ba91f52a93e6a31c0416)
![a, b \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/37993682f4308f970fc0234608a475fc293dc2cc)
![{\ displaystyle (a, b) \ in {\ mbox {Def}} (*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/988dc135ff82a356a8725e612d0768cb79af4eb8)
![{\ displaystyle a * b = c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2aa247b882375ed5430c67bc6f16014fbc44e06)
![{\ displaystyle (a, b) \ in {\ mbox {Def}} (\ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbdcbf86e3879bf82ba1aa4c212d55a7a9cecd5)
![{\ displaystyle a \ circ b = c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/515d82dfb34c61ca424373fc3973632e82e0cd3d)
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ie extension of (Engl. "extension of U" ) when made and followed , and out of following . Example:![{\ boldsymbol U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7400a9c60ebb8eda9fff860d7d8e1435fea8308d)
![a, b \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb00e3fe8849131890d2b4db8373457573bd0a4)
![{\ displaystyle (a, b) \ in {\ mbox {Def}} (*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/988dc135ff82a356a8725e612d0768cb79af4eb8)
![a, b \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/37993682f4308f970fc0234608a475fc293dc2cc)
![{\ displaystyle c \ in M \ land c \ notin U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9ea51c7cc0fd0c4e95f6888aecba76f80d82b2)
![{\ displaystyle \ exists (a, b) \ in U \ times U: (a, b) \ in {\ mbox {Def}} (*) \ land c = a * b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e26cfcfac075e1641a4c172a684b0469fde447)
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- An extension of means full expansion of (Engl. "Complete extension of U" ) if . Example:
![{\ displaystyle {\ boldsymbol {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d5057408f7628386ba91f52a93e6a31c0416)
![{\ boldsymbol U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7400a9c60ebb8eda9fff860d7d8e1435fea8308d)
![{\ boldsymbol U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7400a9c60ebb8eda9fff860d7d8e1435fea8308d)
![{\ displaystyle \ forall (a, b) \ in U \ times U: (a, b) \ in {\ mbox {Def}} (*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77d3e88cf0fb557adce1112a450b4d01832bb024)
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- 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a054d5057408f7628386ba91f52a93e6a31c0416)
![{\ boldsymbol U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7400a9c60ebb8eda9fff860d7d8e1435fea8308d)
![{\ boldsymbol U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7400a9c60ebb8eda9fff860d7d8e1435fea8308d)
![{\ displaystyle (a, b) \ in {\ mbox {Def}} (*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/988dc135ff82a356a8725e612d0768cb79af4eb8)
![{\ displaystyle a * b = c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2aa247b882375ed5430c67bc6f16014fbc44e06)
![{\ displaystyle c \ in U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e74a464b9f106a3d93242b0cb03d55761345294d)
![{\ displaystyle (a, b) \ in {\ mbox {Def}} (\ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbdcbf86e3879bf82ba1aa4c212d55a7a9cecd5)
![{\ displaystyle a \ circ b = c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/515d82dfb34c61ca424373fc3973632e82e0cd3d)
![{\ displaystyle c \ in M \ land c \ notin U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9ea51c7cc0fd0c4e95f6888aecba76f80d82b2)
![{\ displaystyle (a, b) \ in {\ mbox {Def}} (*) \ land (a ', b') \ in {\ mbox {Def}} (*) \ land a * b = a '* b '= c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3ce06a32491f4558d531b4b54119fe250c17db1)
![{\ displaystyle a = a '\ land b = b'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7aea661eae5045ed9fe9d39f48e640de85b8063d)
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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:
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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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c29b0246504cbfa395d4380bc74925d3679aebb)
![{\ displaystyle x, y, z \ in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba283a127121ad64c98d3f69ced0ac4a86ec6414)
![{\ displaystyle x \ circ y \ in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1329b83de7cf2c0f38f3ad2f711618a81bf97f89)
![{\ displaystyle y \ circ z \ in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e052845ee61022ccbb4d056567be276edb26238e)
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exactly when
- and , if (and thus after 1. also )
![{\ displaystyle x \ circ (y \ circ z) = (x \ circ y) \ circ z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40ddaafc638b6d08cd7df41f480db7ffbc5f7561)
![{\ displaystyle x \ circ (y \ circ z) \ in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6d7511bfccf78ee538c6d4c2ef69b2ef729ccd)
![{\ displaystyle (x \ circ y) \ circ z \ in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e918ced10aa8167a8f230eb169772a1d7ed10767)
- A pseudo-magma is commutative if holds for all
![{\ displaystyle (M, \ circ)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c29b0246504cbfa395d4380bc74925d3679aebb)
![x, y \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea304ca242a255b620d3dd16ec47f19efc2e7ab8)
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exactly when
- and , if (and thus after 1. also )
![{\ displaystyle x \ circ y = y \ circ x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2578d2ab8b7ab36781793b5484c664d7f283e93c)
![{\ displaystyle x \ circ y \ in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1329b83de7cf2c0f38f3ad2f711618a81bf97f89)
![{\ displaystyle y \ circ x \ in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc04f1b78534262f7dd2f420b2ce7ee8cccdebd)
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
![\ circ \ colon M \ times M \ to M, (\ operatorname f, \ operatorname g) \ mapsto \ operatorname f \ circ \ operatorname g](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce8405bf05462c076cbeed733b6dc4e47d466bd)
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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.
![\ Sigma ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/807344600a40f1de7136f8b54576e12e9428bef4)
Individual evidence
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^ A b Günther Eisenreich: Lexicon of Algebra . Akademie-Verlag / Springer-Verlag, Berlin 1989, ISBN 3-05-500231-8 .
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↑ 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 .
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^ Yoshifumi Inui, François Le Gall: Quantum Property Testing of Group Solvability . S. 2 , arxiv : 0712.3829 (definition at the beginning of § 2.1).
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↑ 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.