Residual finiteness in permutation varieties of semigroups
A semigroup S is called residually finite if for all \(a\neq b\) in S there is a congruence \(\rho\) on S such that S/\(\rho\) is finite and \(\neg (a \rho b)\). In this paper the permutation varieties of semigroups, in which every finitely generated (f.g.) member is residually finite, are determined. Based on theorems of Yu. Mal'tsev, A. Ananin and T. Evans on varieties of associative algebras the following theorem is proved: If E is a set of permutation identities and if \(V_ E (M_ E)\) denotes the variety of semigroups (associative algebras) then the following are equivalent: 1) \(V_ E\) satisfies \(xy^ nx=y^ iy^{n-1-j}xy^ j\) for some \(i,j,n\in {\mathbb{Z}}^{+,0}\) with \(0<i+j\leq n\); 2) \(V_ E\) is locally representable, i.e. every f.g. member of \(V_ E\) can be embedded into a matrix algebra over a field of characteristic zero; 3) \(V_ E\) is locally Hopfian, i.e. no f.g. member of \(V_ E\) is isomorphic to a proper image; 4) \(V_ E\) is locally residually finite, i.e. every f.g. member of \(V_ E\) is residually finite; 5) \(V_ E\) is locally Noetherian, i.e. the congruence lattice of any f.g. member of \(V_ E\) satisfies the ascending chain condition; 6) \(M_ E\) is locally Noetherian; 7) Every f.g. member of \(V_ E\) is finitely presented. - Finally, the solvability of the word problem for any finite presentation of an f.g. semigroup S in \(V_ E\) satisfying (1) and thus any condition above, is proved.
scientific article; zbMATH DE number 3879163
scientific article