Normally Ordered Semigroups

1992

Let X be a totally ordered set and consider the semigroups of orderdecreasing(increasing) full (partial, partial one-to-one) transformations of X. In this Thesis the study of order-increasing full (partial, partial one-to-one) transformations has been reduced to that of order-decreasing full (partial, partial one-to-one) transformations and the study of order-decreasing partial transformations to that of order-decreasing full transformations for both the finite and infinite cases. For the finite order-decreasing full (partial one-to-one) transformation semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden (1990) concerning products of idempotents (quasi-idempotents), and concerning combinatorial and rank properties. By contrast with the semigroups of order-preserving transformations and the full transformation semigroup, the semigroups of orderdecreasing full (partial one-to-one) transformations and their Rees quotient semigroups are not regular. They are, however, abundant (type A) semigroups in the sense of Fountain (1982,1979). An explicit characterisation of the minimum semilattice congruence on the finite semigroups of order-decreasing transformations and their Rees quotient semigroups is obtained. If X is an infinite chain then the semigroup S of order-decreasing full transformations need not be abundant. A necessary and sufficient condition on X is obtained for S to be abundant. By contrast, for every chain X the semigroup of order-decreasing partial one-to-one transformations is type A. The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing full (partial one-to-one) transformations have been investigated. vt GLOSSARY OF NOTATION <A> the semigroup generated by A A* the upper saturation of A NO the smallest infinite cardinal AQE(S) the set of amenable elements of S Bn the nth Bell's exponential number A Green's relation defined by Z join

2012

For n ∈ IN, let Xn = {1 &lt; 2 &lt; · · · &lt; n} be a finite chain with n elements. As usual, we denote by PTn the semigroup of all partial transformations α : Xn → Xn under composition. A transformation α ∈ PTn is called order-preserving if x ≤ y implies xα ≤ yα for all x, y from the domain of α. As usual, POn denotes the subsemigroup of PTn of all partial order-preserving transformations of Xn. This semigroup has been extensively studied. In recent years, interest in maximal subsemigroups of the transformation semigroups arises. In particular, Xiuliang Yang [6] characterized the maximal subsemigroups of the semigroup On of all full order-preserving transformations. Dimitrova and Koppitz [1] classified the maximal subsemigroups of the ideals of On. Ganyushkin and Mazorchuk [3] gave a description of the maximal subsemigroups of the semigroup POIn of all partial order-preserving injections. In [2], Dimitrova and Koppitz characterized the maximal subsemigroups of the ideals of the se...

2015

A b s t r ac t. It is well-known [16] that the semigroup T n of all total transformations of a given n-element set X n is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of order-preserving transformations on a finite chain X n is a semilattice of idempotents, and so the semigroup of all order-preserving transformations of X n is not covered by its inverse subsemigroups. This result is used to show that the semigroup of all orientation-preserving transformations and the semigroup of all orientation-preserving or orientation-reversing transformations of the chain X n are covered by their inverse subsemigroups precisely when n 3.

Semigroup Forum, 1985

arXiv (Cornell University), 2017

A regular ordered semigroup S is called right inverse if every principal left ideal of S is generated by an R-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element a ∈ S are R-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.

2015

It is well-known [16] that the semigroup \(\mathcal{T}_n\) of all total transformations of a given \(n\)-element set \(X_n\) is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of order-preserving transformations on a finite chain \(X_n\) is a semilattice of idempotents, and so the semigroup of all order-preserving transformations of \(X_n\) is not covered by its inverse subsemigroups. This result is used to show that the semigroup of all orientation-preserving transformations and the semigroup of all orientation-preserving or orientation-reversing transformations of the chain \(X_n\) are covered by their inverse subsemigroups precisely when \(n \leq 3\).

Glasgow Mathematical Journal, 1990

We make the convention that if a is an element of a semigroup Q then by writing a–1 it is implicit that a lies in a subgroup of Q and has inverse a–1 in this subgroup; equivalently, a ℋ a2 and a–1 is the inverse of a in Ha.A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as a–1b where a, b ∈ S and, in addition, every element of S satisfying a weak cancellation condition which we call square-cancellable lies in a subgroup of Q. The notions of right order and semigroup of right quotients are defined dually; if S is both a left order and a right order in Q then S is an order in Q and Q is a semigroup of quotients of S.

2009

Semilattice-ordered semigroup is important algebraic structure. It is equipped with the natural defined positive quasi-antiorder relation.

Semigroup Forum, 1996

Semilattice-ordered semigroup is an important algebraic structure. It is ordered semigroup under anti-order. Some basic properties of semillatice-ordered semigroups with apartness are given by constructive point of view. Let I and K be compatible an ideal and an anti-ideal of semilattice-ordered semigroup S. Constructions of compatible congruence E(I) and anti-congruence Q(K) on S, generated by I and K respectively, are given. Besides, we construct compatible order T and anti-order θ T on factor-semigroup S/(E(I), Q(K)). Some basic properties of such constructed semigroups are given.