DECIDING SOME EMBEDDABILITY PROBLEMS FOR SEMIGROUPS OF MAPPINGS

Abstract

This note, based on a part on the talk of the second author at the Braga Conference, gives a method allowing solution of decidability problems including the ones mentioned in the abstract. These are not open problems but we believe that the simple technique used here is worthy of attention. The problem of embeddability into an inverse semigroup was solved by Schein nearly forty years ago [1]. Schein showed that a necessary and sufficient condition for embeddability into an inverse semigroup was that the so called strong quasi-order relation on the semigroup be an order relation. This can be reformulated as an infinite system of quasi-identites (equational implications) which, although containing redundancies, cannot be replaced by any finite system. It does also furnish an effective procedure for deciding the embeddability question for any finite semigroup. The corresponding question for semigroups of isotone mappings has more recently received attention as semigroups related to semigroups of orderpreserving mappings have been intensively studied. Here the related papers [2] of Vernitskil and [3] of Volkov in the Proceedings of St Andrews Conference on Semigroups in 1997 are relevant. In particular Lemma 5.1 of [3] shows that the question of whether a finite semigroup can be faithfully represented as a semigroup of partial order-preserving mappings on a chain is decidable. Volkov presented further results along these lines at the GAP Conference in Lisbon in 1997 which included the case of embeddability into the semigroups On of total order-preserving mappings on a finite chain of length n and there is continuing joint work of RepnitskiY Volkov and VernitskiY on bases for quasivarieties generated by isotone mappings.