Residually-finite semi-group

finitely-approximable semi-group

A semi-group for any two distinct elements and of which there is a homomorphism of it into a finite semi-group such that . The property of a semi-group being residually finite is equivalent to that of 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 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 , 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 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 -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 by an arbitrary residually-finite semi-group is residually-finite if, for example, is reductive, that is, if any two distinct elements of induce distinct left and distinct right inner translations; in particular, if 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 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 -simple semi-groups with a finite number of - or -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 , , , and be, respectively, the two-element left zero and right zero semi-groups, the semi-group with zero multiplication, and a semi-lattice, let be the three-element semi-group , where , and the remaining products are equal to , and let be the semi-group anti-isomorphic to . A variety consists of residually-finite semi-groups if and only if is generated by a subset of one of the following three sets: , , , where is a finite group with Abelian Sylow subgroups and 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)