The Gap Between Partial and Full

2012

Let P Tn be the semigroup of all partial transformations on an n -element set. A transformation α ∈ P Tn is called order-preserving if x ≤ y implies xα ≤ yα for all x, y from the domain of α. In this paper we describe the maximal subsemigroups of the semigroup P On of all partial order-preserving transformations. * 2000 Mathematics Subject Classification: 20M20.

Monatshefte fur Mathematik, 2003

In 1986, Kowol and Mitsch studied properties of the so-called 'natural partial order' ≤ on T (X), the total transformation semigroup defined on a set X. In particular, they determined when two total transformations are related under this order, and they described the minimal and maximal elements of (T (X), ≤). In this paper, we extend that work to the semigroup P (X) of all partial transformations of X, compare ≤ with another 'natural' partial order on P (X), characterise the meet and join of these two orders, and determine the minimal and maximal elements of P (X) with respect to each order.

Ukrainian Mathematical Journal, 2021

In this paper, we determine the relative rank of the semigroup T (X, Y) of all transformations on a finite chain X with restricted range Y ⊆ X modulo the set OP(X, Y) of all orientation-preserving transformation in T (X, Y). Moreover, we state the relative rank of the semigroup OP(X, Y) modulo the set O(X, Y) of all order-preserving transformations in OP(X, Y). In both cases we characterize the minimal relative generating sets.

arXiv: Rings and Algebras, 2021

Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X, Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X, Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size.

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

Asian-European Journal of Mathematics, 2020

In this paper, we determine the relative rank of the semigroup [Formula: see text] of all orientation-preserving transformations on infinite chains modulo the semigroup [Formula: see text] of all order-preserving transformations.

Semigroup Forum, 1985

Semigroup Forum, 2013

Let X be a finite or infinite chain and let O(X) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of O(X) and Green's relations on O(X). In fact, more generally, if Y is a nonempty subset of X and O(X, Y ) the subsemigroup of O(X) of all elements with range contained in Y , we characterize the largest regular subsemigroup of O(X, Y ) and Green's relations on O(X, Y ). Moreover, for finite chains, we determine when two semigroups of the type O(X, Y ) are isomorphic and calculate their ranks. 2000 Mathematics subject classification: 20M20, 20M10.

Journal of Algebra, 2004

Let PC n be the semigroup of all decreasing and order-preserving partial transformations of a finite chain. It is shown that |PC n | = r n , where r n is the large (or double) Schröder number. Moreover, the total number of idempotents of PC n is shown to be (3 n + 1)/2.

2003

Let X be an arbitrary poset. The semigroup I + (X ) of all order-increasing partial oneone mappings of X is shown to be an ample semigroup with some interesting properties. Moreover, a necessary and sufficient condition (on the totally ordered sets X and Y ) for I + (X ) and I + (Y ) to be isomorphic has been established.