Elementary orthodox semigroups

Bisimple monogenic orthodox semigroups

“…The purpose of this paper is to determine the structure of all semigroups of that class using only some standard facts about orthodox semigroups. Partial results in this direction, based on the study of the lattice of congruences on the free orthodox semigroup generated by a pair of mutually inverse elements, can be found in [5] -see Section 1 below for more details.…”

“…In [4,5], a semigroup S was termed elementary if S = A ∪ B for two nonempty subsets A and B of S such that a ⊥ b for all a ∈ A and b ∈ B, and an orthodox semigroup S = a, b with a ⊥ b was referred to as "an elementary orthodox semigroup on two mutually inverse generators". This notion was introduced as a generalization of (and by analogy with) "elementary generalized groups" studied by Gluskin in [6].…”

“…In view of the above, it is natural to ask: If S = x, y is a semigroup such that x ⊥ y, under what conditions is S orthodox? The answer follows easily from [5,Theorem 1.1], stating that the free monogenic orthodox semigroup FO has the following presentation: FO = p, q | p n ⊥ q n for all n ∈ N ; we will also denote this semigroup by FO(p, q) if we wish to name the generators explicitly. Lemma 1.1.…”

“…In [5,Section II], using the description of FO from [5, Section I] and a number of results from [3] and a few other papers, Eberhart and Williams analyzed the lattice of congruences on FO and constructed several interesting monogenic orthodox semigroups. In particular, in [5,Results 2.3,2.4] they exhibited the following two infinitely presented bisimple monogenic orthodox semigroups: FO/α ′ = p, q | p n ⊥ q n (∀ n ∈ N) and pq = p 2 q 2 and FO/σ ′ = p, q | p n ⊥ q n (∀ n ∈ N) and pq = p 2 q 2 , qp = q 2 p 2 , where α ′ and σ ′ are certain congruences on FO defined in [5] in several steps using the description of congruences on the free monogenic inverse semigroup and the results about the lattice of congruences on FO (the interested reader is referred to [5] for definitions of α ′ and σ ′ ; they will not be needed in this paper). It is immediate from Lemma 1.1 that FO/α ′ and FO/σ ′ can be finitely presented:…”

“…The semigroups FO/α ′ and FO/σ ′ given in [5,Results 2.3 and 2.4], respectively, have the following finite presentations:…”