Equational theories of semigroups with enriched signature
Karl Auinger, Igor Dolinka, Mikhail Vladimirovich Volkov · arXiv (Cornell University) · 2009
Abstract. We present sufficient conditions for a unary semigroup variety to have no finite basis for its equational theory. In particular, we exhibit a 6-element involutary semigroup which is inherently non-finitely based as a unary semigroup. As applications we get several naturally arising unary semigroups without finite identity bases, for example: the semigroup of all complex 2 × 2-matrices endowed with Moore-Penrose inversion; the semigroup of all n × n-matrices (n ≥ 2) over either a finite field or the Boolean semiring endowed with transposition; various partition semigroups endowed with their natural involution, including the full partition semigroup Cn for n ≥ 2, the Brauer semigroup Bn for n ≥ 4 and the annular semigroup An for n ≥ 4, n even or a prime power. We also show that similar techniques apply to the finite basis problem for existence varieties of locally inverse semigroups. Contents