Algorithmic problems in varieties of semigroups (Q1824703)

Let \({\mathcal M}\) be a finitely based nonperiodic variety of semigroups, then the following conditions are equivalent: (1) All the finitely presented semigroups (f.p.s.) of \({\mathcal M}\) have solvable word problem. (2) All the f.p.s. of \({\mathcal M}\) have solvable elementary theory. (3) All the f.p.s. of \({\mathcal M}\) are finitely approximated. (4) All the f.p.s. of \({\mathcal M}\) are imbeddable in matrix semigroups (over a field). (5) \({\mathcal M}\) has identities of the form \(x^ ny(z^ kt^ k)^ pz^ m=x^ m(t^ kx^ k)^ pyz^ n\), \(xy^ nz=y^ kzy^ mzy^ p\), \(n>m.\) (6) \({\mathcal M}=\{\bar P\times P^ 1\), \(\bar P^ 1\times P\), \(T\}\), where \(P=\{e_{11},e_{12},0\}\) \(T=\{e_{11},e_{12},e_{22},0\}\), \({}^- \) is an antiisomorphism, \(P^ 1=P\cup \{1\}.\)