Residually-finite semi-group

finitely-approximable semi-group

A 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); it is closely connected with algorithmic problems (cf. 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 [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) [4]; see also Residually-finite group.

The direct product, the free product, the ordinal sum (see 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 [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). 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 [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

[1]	A.I. Mal'tsev, "Homomorphisms onto finite groups" Uchen. Zap. Ivanovsk. Ped. Inst. , 18 (1958) pp. 49–60 (In Russian)
[2]	E.A. Golubov, "Finitely approximable regular semi-groups" Math. Notes , 17 : 3 (1975) pp. 247–251 Mat. Zam. , 17 : 3 (1975) pp. 423–432
[3]	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
[4]	G. Lallement, "On nilpotency and residual finiteness in semigroups" Pacific J. Math. , 42 : 3 (1972) pp. 693–700

References

[a1]	P.M. Cohn, "Universal algebra" , Reidel (1981)