The proof and the generalization of Higgins' theorem of divisors of semigroups of order-preserving mappings
The main result of this paper is that every finite semigroup \(S\) the idempotents of which form an ideal is isotone, in the sense that \(S\) is a homomorphic image of a semigroup of order-preserving transformations. This generalizes a result of the reviewer [Int. J. Algebra Comput. 5, No. 6, 725-742 (1995; Zbl 0842.20054)] that any finite band is isotone, a fact there proved by implementation of the solution of the word problem for free bands by isotone automata. The proof of the authors is shorter and is based on construction of \(S_\ell\), a semigroup which extends \(S\) by adjoining a copy of the set \(S^1\) to act as a left zero ideal in a certain fashion. It transpires that if \(S\) is isotone then so is \(S_\ell\). This, together with the self-duality of the pseudovariety \(\mathcal O\) of all semigroups generated by semigroups of order-preserving mappings, allows the authors to devise a contradiction argument based on Gerard's solution to the word problem for semigroups satisfying the identity \(x_1x_2\dots x_n=(x_1x_2\dots x_n)^2\).
semigroup of order-preserving transformations
scientific article; zbMATH DE number 862276