Finite orthodox locally idempotent semigroups having no finite basis of biidentities.
This paper provides new deep results in the theory of orthodox semigroups. For a variety \(\mathcal A\) of bands, denote by \(\mathcal{AOLB}\) the e-variety of all orthodox, locally idempotent semigroups whose band of idempotents is in \(\mathcal A\). The first fundamental result is the solution to the word problem for the bifree objects in \(\mathcal{AOLB}\). Next, a sub-e-variety \(\mathcal{OZ}\) of \((\mathcal{LRB})\mathcal{OLB}\) (here \(\mathcal{LRB}\) denotes the variety of all left regular bands) is constructed which is (i) finitely generated and (ii) not finitely based. As a consequence, the bifree object (on at least seven generators) in \(\mathcal{OZ}\) serves as an example of a finite orthodox, locally idempotent semigroup which has no finite basis for its bi-identities nor for its semigroup identities -- this solves a longstanding open problem in the theory of identity bases [see \textit{L. N. Shevrin} and \textit{M. V. Volkov}, Izv. Vyssh. Uchebn. Zaved., Mat. 1985, No. 11(282), 3-47 (1985; Zbl 0629.20029) and \textit{M. V. Volkov}, Sci. Math. Jpn. 53, No. 1, 171-199 (2001; Zbl 0990.20039)]; note that the Brandt monoid \(B_2^1\) is \textit{not} a member of the variety generated by this object. On the other hand it is pointed out that \((\mathcal{LRB})\mathcal{OLB}\) itself is also finitely generated, thereby providing an example of a finite orthodox semigroup which is neither strict nor completely regular and which has a finite basis for its bi-identities. The methods of the paper are syntactical; the main ingredients are the invention and the clever use of some, quite sophisticated, new combinatorial invariants of words on a doubled alphabet \(X\cup X'\).
scientific article; zbMATH DE number 1975851
scientific article