Introduction to semigroup theory

Abstract In this paper, we study the ordered regular equivalence relations on ordered semihypergroups in detail. To begin with, we introduce the concept of weak pseudoorders on an ordered semihypergroup, and investigate several related properties. In particular, we construct an ordered regular equivalence relation on an ordered semihypergroup by a weak pseudoorder. As an application of the above result, we completely solve the open problem on ordered semihypergroups introduced in [B. Davvaz, P. Corsini and T. Changphas, Relationship between ordered semihypergroups and ordered semigroups by using pseuoorders, European J. Combinatorics 44 (2015), 208–217]. Furthermore, we establish the relationships between ordered regular equivalence relations and weak pseudoorders on an ordered semihypergroup, and give some homomorphism theorems of ordered semihypergroups, which are generalizations of similar results in ordered semigroups.

A further study on ordered regular equivalence relations in ordered semihypergroups

The aim of this paper is to study the varieties of semilattice-ordered Burnside semigroups satisfying $$x^3\approx x$$
 and $$xy\approx yx.$$
 It is shown that the collection of all such varieties forms a distributive lattice of order 9. Also, all of them are finitely based and finitely generated. This gives a generalization and expansion of the results obtained by McKenzie and Romanowska (Contrib Gen Algebra Proc Klagenf Conf 1978 1:213–218, 1979).

The varieties of semilattice-ordered semigroups satisfying $$x^3\approx x$$ and $$xy\approx yx$$xy≈yx

The aim of this paper is to study the varieties of ai-semirings satisfying ${x^{3}\approx x}$ . It is shown that the collection of all such varieties forms a distributive lattice of order 179. Also, all of them are finitely based and finitely generated. This generalizes and extends the main results obtained by Ghosh et al., Pastijn and Ren and Zhao.

On the varieties of ai-semirings satisfying ${x^{3}\approx x}$

We establish that, under certain closure assumptions on a pseudovariety of semigroups, the corresponding relatively free profinite semigroups freely generated by a non-singleton finite set act faithfully on their minimum ideals. As applications, we enlarge the scope of several previous join irreducibility results for pseudovarieties of semigroups, which turn out to be even join irreducible in the lattice of pseudovarieties of ordered semigroups, so that, in particular, they are not generated by proper subpseudovarieties of ordered semigroups. We also prove the stronger form of join irreducibility for the Krohn-Rhodes complexity pseudovarieties, thereby solving a problem proposed by Rhodes and Steinberg.

Representations of relatively free profinite semigroups, irreducibility, and order primitivity

Let B(k,0,n) denote the group with k generators which is free in the group variety defined by the identity xn=1. Let Bslo(k,1,n) denote the semilattice-ordered semigroup with k generators which is free in the semilattice-ordered semigroup variety defined by the identity xn=x. We prove a generalization of the Green-Rees theorem: Bslo(k,1,n) is finite for all k≥1 if and only if B(k,0,n−1) is finite for all k≥1. We find a formula for card(Bslo(1,1,n)). We construct Bslo(k,1,n) for some concrete values of k and n.

On free semilattice-ordered semigroups satisfying xn=x

We use syntactical methods (fully invariant stable quasiorders) to study varieties of ordered semigroups. We prove that the lattice of all varieties of semigroups is embedded into the lattice of all varieties of ordered semigroups. The lattice of all varieties of ordered bands is completely described.

On varieties of ordered semigroups

We define semantically a partial multiplication on the lattice of all e–varieties of regular semigroups. In the case that the first factor is an e–variety of orthodox semigroups we describe our multiplication syntactically in terms of biinvariant congruences.

A multiplication of $e$-varieties of orthodox semigroups

A multiplication of e-varieties of regular $E$-solid semigroups by inverse semigroup varieties is described both semantically and syntactically. The associativity of the multiplication is also proved.

A multiplication of e-varieties of regular $E$-solid semigroups by inverse semigroup varieties

We construct, classify and enumerate all non-isomorphic ordered semigroups of size at most 7. In the special case of linearly ordered semigroups we achieve size 10.

Martin Kuřil

Papers

On free semilattice-ordered semigroups satisfying xn=x

On varieties of ordered semigroups

A multiplication of $e$-varieties of orthodox semigroups

A multiplication of e-varieties of regular $E$-solid semigroups by inverse semigroup varieties

Ordered semigroups of size at most 7 and linearly ordered semigroups of size at most 10