View source for Residually-finite semi-group

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''finitely-approximable semi-group''

A [[Semi-group|semi-group]] for any two distinct elements $a$ and $b$ of which there is a homomorphism $\phi$ of it into a finite semi-group $S$ such that $\phi(a)\neq\phi(b)$. The property of a semi-group $S$ being residually finite is equivalent to that of $S$ being a subdirect product of finite semi-groups. Residual finiteness is one of the more important finiteness conditions (see [[Semi-group with a finiteness condition|Semi-group with a finiteness condition]]); it is closely connected with algorithmic problems (cf. [[Algorithmic problem|Algorithmic problem]]): if $S$ is a finitely-presented residually-finite semi-group, then there is an algorithm for solving the word problem in it. The residually-finite semi-groups include the free semi-groups, the free commutative semi-groups, the free nilpotent semi-groups of class $n$, the free inverse semi-groups (as algebras with two operations), the semi-lattices, the finitely-generated commutative semi-groups [[#References|[1]]], the finitely-generated semi-groups of matrices over a nilpotent or commutative ring, and the finitely-generated regular semi-groups that are nilpotent of class $n$ in the sense of Mal'tsev (see [[Nilpotent semi-group|Nilpotent semi-group]]) [[#References|[4]]]; see also [[Residually-finite group|Residually-finite group]].

The [[Direct product|direct product]], the [[Free product|free product]], the ordinal sum (see [[Band of semi-groups|Band of semi-groups]]), and the $0$-direct union of an arbitrary set of residually-finite semi-groups are also residually-finite semi-groups. Other constructions do not, generally speaking, preserve residual finiteness. An ideal extension of a residually-finite semi-group $S$ by an arbitrary residually-finite semi-group is residually-finite if, for example, $S$ is reductive, that is, if any two distinct elements of $S$ induce distinct left and distinct right inner translations; in particular, if $S$ is a cancellation or inverse semi-group. The semi-lattice of a family of reductive residually-finite semi-groups is a residually-finite semi-group.

If $S$ is a residually-finite semi-group, then all maximal subgroups of it are residually finite. For certain types of semi-groups this necessary condition is also sufficient; such as: regular semi-groups with a finite number of idempotents in every principal factor [[#References|[2]]], Clifford inverse semi-groups, and completely $0$-simple semi-groups with a finite number of $\mathcal L$- or $\mathcal R$-classes (see [[Green equivalence relations|Green equivalence relations]]). For a number of classes of semi-groups a characterization of the residually-finite semi-groups in them has been obtained in terms not using reduction to maximal subgroups.

Varieties of residually-finite semi-groups have been characterized in several ways [[#References|[3]]]. One such is the following. Let $L$, $R$, $N$, and $I$ be, respectively, the two-element left zero and right zero semi-groups, the semi-group with zero multiplication, and a semi-lattice, let $P$ be the three-element semi-group $\{e,p,0\}$, where $e^2=e$, $ep=p$ and the remaining products are equal to $0$, and let $P^*$ be the semi-group anti-isomorphic to $P$. A variety $M$ consists of residually-finite semi-groups if and only if $M$ is generated by a subset of one of the following three sets: $\{L,R,N,I,G\}$, $\{R,P,C\}$, $\{L,P^*,C\}$, where $G$ is a finite group with Abelian Sylow subgroups and $C$ is a finite cyclic group.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,   "Homomorphisms onto finite groups"  ''Uchen. Zap. Ivanovsk. Ped. Inst.'' , '''18'''  (1958)  pp. 49–60  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.A. Golubov,   "Finitely approximable regular semi-groups"  ''Math. Notes'' , '''17''' :  3  (1975)  pp. 247–251  ''Mat. Zam.'' , '''17''' :  3  (1975)  pp. 423–432</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Golubov,   M.V. Sapir,   "Varieties of finitely approximable semigroups"  ''Soviet Math. Dokl.'' , '''20''' :  4  (1979)  pp. 828–832  ''Dokl. Akad. Nauk SSSR'' , '''247''' :  5  (1979)  pp. 1037–1041</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Lallement,   "On nilpotency and residual finiteness in semigroups"  ''Pacific J. Math.'' , '''42''' :  3  (1972)  pp. 693–700</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR></table>