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AbstractFor n∈Z+ , let Xn={1,…,n} be a finite chain and Tn be the full transformation semigroup on Xn . If OCTn is the set of all order-preserving and full contraction mappings on Xn , then it is well-known that OCTn is a submonoid of Tn . In this paper we give a monoid presentation of OCTn . [ABSTRACT FROM AUTHOR]

Let I n be the symmetric inverse semigroup on X n = { 1 , 2 , ⋯ , n } , and let O D P n be the subsemigroup of I n consisting of all order-preserving partial isometries of X n . Let Y = { 1 , 2 , ⋯ , r } be a non-empty subset of X n . Define O D P I n , r = { α ∈ O D P n : im α ⊆ Y }. In this paper, we give a necessary and sufficient condition for O D P I n , r to be an inverse semigroup and characterize its Green relations. Moreover, the cardinality and the rank of O D P I n , r are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of an orientation-preserving transformation on an arbitrary chain, as a natural extension for infinite chains of the well known concept for finite chains introduced in 1998 by McAlister and, independently, in 1999 by Catarino and Higgins. We consider the monoid POP (X) (X) of all orientation-preserving partial transformations on a finite or infinite chain X and its submonoids and POPI (X) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on X, respectively. The monoid OPPO (X) of all order-preserving partial transformations on X and its injective counterpart POI (X) are also considered. We study the regularity and give descriptions of the Green's relations of the monoids POP (X) , PO (X) , OP (X) , POPI (X) , and POI (X). [ABSTRACT FROM AUTHOR]

It is well known that the semigroup T(X) of all transformations on a set X is regular, but its subsemigroups do not need to be. Consider an ordered set (X ; ≤) whose order forms a path with alternating orientation. The semigroup OT (X) of all order-preserving transformations on (X ; ≤) is a subsemigroup of T(X). Some characterizations of regular semigroups OT (X) were given. For a non-empty subset S of X, we define the set OT S (X) = { α ∈ OT (X) ∣ ran α = S }. In this paper, we show that in general, OT S (X) is not a subsemigroup of T(X). A necessary and sufficient condition for OT S (X) to be a regular subsemigroup of T(X) is given. Some regular subsemigroups of T(X) that are contained in OT S (X) are studied. We investigate the left (right) regularity of OT (X) . We obtain that left regular and right regular elements in OT (X) are identical. A characterization of left (right) elements in OT (X) is given. We classify all left (right) regular semigroups OT (X) . [ABSTRACT FROM AUTHOR]

A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup TF n of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of TF n and provide a minimal generating set for TF n . Moreover, a formula for the number of idempotents in TF n is given. [ABSTRACT FROM AUTHOR]

For n ∈ N , let O n be the semigroup of all order-preserving transformations on the finite chain X n = { 1 , ... , n } , under its natural order. For any non-empty subset A of X n , let O n (A) and O n + (A) be the subsemigroups of all order-preserving and A-decreasing, and of all order-preserving and A-increasing transformations on X n , respectively. In this paper we obtain formulae for the number of elements and for the number of idempotents in O n (A) . Moreover, we show that O n (A) contains a zero element if and only if 1 ∈ A , and then we obtain the number of nilpotents in O n (A) when 1 ∈ A . [ABSTRACT FROM AUTHOR]

In this paper we consider various classes of monoids of transformations of a finite chain, including those of transformations that preserve or reverse either the order or the orientation. In line with Howie and McFadden [24], we complete the study of the ranks (and of idempotent ranks, when applicable) of all their ideals. [ABSTRACT FROM AUTHOR]

In this paper we calculate presentations for same natural monoids of transformations on a chain Xn &lcub;1 < 2 < ... < n&rcub;. First we consider OD, [PODn], the monoid of all full [partial] transformations on Xn that preserve or reverse the order. Two other monoids of partial transformations on Xn we took at are PODn and PORn—the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation. [ABSTRACT FROM AUTHOR]