On orientation-preserving transformations of a chain

Abstract

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 [27] and, independently, in 1999 by Catarino and Higgins [8]. We consider the monoid POP(X) of all orientation-preserving partial transformations on a finite or infinite chain X and its submonoids OP(X) and POPI(X) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on X, respectively. The monoid PO(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).

On relative ranks of the semigroup of orientation-preserving transformations on infinite chains

“…A generalization of the concept of an order-preserving transformation is the concept of an orientation-preserving transformation, which was introduced in 1998 by McAlister [15] and, independently, one year later by Catarino and Higgins [1], but only for finite chains. In [8], Fernandes, Jesus, and Singha introduced the concept of an orientation-preserving transformation on an infinite chain. It generalizes the concept for a finite chain.…”

“…Definition 1 [8] Let α ∈ T (X). We say that α is an orientation-preserving transformation if there exists a non-empty subset X 1 of X such that:…”

“…Denote by OP(X) the subset of T (X) of all orientation-preserving transformations. In [8], Fernandes, Jesus, and Singha proved that OP(X) is a monoid. Moreover, they proved that if α ∈ OP(X) is a non-constant transformation then α admits a unique ideal.…”

On Relative Ranks of the Semigroup of Orientation-preserving Transformations on Infinite Chains

“…A generalization of the concept of an order-preserving transformation is the concept of an orientation-preserving transformation, which was introduced in 1998 by McAlister [19] and, independently, one year later by Catarino and Higgins [3], but only for finite chains. In [8], Fernandes, Jesus, and Singha introduced the concept of an orientation-preserving transformation on an infinite chain. It generalizes the concept for a finite chain.…”

“…Definition 1 [8] Let α ∈ T (X). We say that α is an orientation-preserving transformation if there exists a non-empty subset X 1 of X such that: (1) α is order-preserving both on X 1 and on X 2 = X \ X 1 ;…”