Difference between revisions of "Imbedding of semi-groups"
From Encyclopedia of Mathematics
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| − | A [[Monomorphism|monomorphism]] of a [[Semi-group|semi-group]] into a [[Group|group]]. A semi-group | + | A [[Monomorphism|monomorphism]] of a [[Semi-group|semi-group]] into a [[Group|group]]. A semi-group $S$ is imbeddable in a group $G$ if $S$ is isomorphic to a sub-semi-group of $G$. Necessary and sufficient conditions for imbeddability of a semi-group into a group were found by A.I. Mal'tsev [[#References|[1]]] (see also [[#References|[3]]]). These conditions form an infinite system of conditional identities (or quasi-identities, cf. [[Quasi-identity|Quasi-identity]]), in particular, the following ones: |
| − | + | $$ap=aq\to p=q,\quad pa=qa\to p=q$$ | |
(cancellation laws); | (cancellation laws); | ||
| − | + | $$ap=bq,ar=bs,cp=dq\to cr=ds,$$ | |
| − | where | + | where $a,b,c,d,p,q,r,s$ are elements of the semi-group. The class of semi-groups imbeddable in groups cannot be characterized by a finite system of conditional identities [[#References|[2]]]. A number of sufficient conditions for imbeddability of a semi-group into a group are known. The most important ones are the following. If $S$ is a semi-group with cancellation and if for any elements $a,b$ of $S$ there exist elements $x,y\in S$ such that $ax=by$ (Ore's condition), then $S$ is imbeddable in a group. If $S$ is a semi-group with cancellation in which it always follows from the equality $ab=cd$ that either $a=cx$ or $c=ax$ for some element $x\in S$, then $S$ is imbeddable in a group [[#References|[4]]]. Sufficient conditions for imbeddability, formulated in the language of graph theory (cf., for example, [[#References|[5]]]), are known. |
====References==== | ====References==== | ||
Revision as of 10:12, 19 August 2014
into groups
A monomorphism of a semi-group into a group. A semi-group $S$ is imbeddable in a group $G$ if $S$ is isomorphic to a sub-semi-group of $G$. Necessary and sufficient conditions for imbeddability of a semi-group into a group were found by A.I. Mal'tsev [1] (see also [3]). These conditions form an infinite system of conditional identities (or quasi-identities, cf. Quasi-identity), in particular, the following ones:
$$ap=aq\to p=q,\quad pa=qa\to p=q$$
(cancellation laws);
$$ap=bq,ar=bs,cp=dq\to cr=ds,$$
where $a,b,c,d,p,q,r,s$ are elements of the semi-group. The class of semi-groups imbeddable in groups cannot be characterized by a finite system of conditional identities [2]. A number of sufficient conditions for imbeddability of a semi-group into a group are known. The most important ones are the following. If $S$ is a semi-group with cancellation and if for any elements $a,b$ of $S$ there exist elements $x,y\in S$ such that $ax=by$ (Ore's condition), then $S$ is imbeddable in a group. If $S$ is a semi-group with cancellation in which it always follows from the equality $ab=cd$ that either $a=cx$ or $c=ax$ for some element $x\in S$, then $S$ is imbeddable in a group [4]. Sufficient conditions for imbeddability, formulated in the language of graph theory (cf., for example, [5]), are known.
References
| [1] | A.I. Mal'tsev, "On inclusion of associative systems in a group" Mat. Sb. , 6 (48) : 2 (1939) pp. 331–336 (In Russian) (German abstract) |
| [2] | A.I. Mal'tsev, "On inclusion of associative systems in a group II" Mat. Sb. , 8 (50) : 2 (1940) pp. 251–264 (In Russian) |
| [3] | P.M. Cohn, "Universal algebra" , Reidel (1981) |
| [4] | R. Doss, "Sur l'immersion d'une semi-groupe dans une groupe" Bull. Sci. Math. (2) , 72 (1948) pp. 139–150 |
| [5] | S.I. [S.I. Adyan] Adjan, "Defining relations and algorithmic problems for groups and semigroups" Proc. Steklov Inst. Math. , 85 (1967) Trudy Mat. Inst. Steklov. , 85 (1966) |
Comments
Instead of imbedding the word embedding is also used.
References
| [a1] | A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) |
How to Cite This Entry:
Imbedding of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_of_semi-groups&oldid=32995
Imbedding of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_of_semi-groups&oldid=32995
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article