TY - JOUR

T1 - Regular elements and Green's relations in generalized transformation semigroups

AU - Mendes-Gonçalves, S.

AU - Sullivan, Robert

AU - Gould, V.

PY - 2013

Y1 - 2013

N2 - If X and Y are sets, we let P(X, Y) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). If θ ∈ P(Y, X), then P(X, Y) is a so-called "generalized semigroup" of transformations under the "sandwich operation": α * β = α θ β, for each α, β ∈ P(X, Y). We denote this semigroup by P(X, Y, θ) and, in this paper, we characterize Green's relations on it: that is, we study equivalence relations which determine when principal left (or right, or 2-sided) ideals in P(X, Y, θ) are equal. This solves a problem raised by Magill and Subbiah in 1975. We also discuss the same idea for important subsemigroups of P(X, Y, θ) and characterize when these semigroups satisfy certain regularity conditions. © 2013 World Scientific Publishing Company.

AB - If X and Y are sets, we let P(X, Y) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). If θ ∈ P(Y, X), then P(X, Y) is a so-called "generalized semigroup" of transformations under the "sandwich operation": α * β = α θ β, for each α, β ∈ P(X, Y). We denote this semigroup by P(X, Y, θ) and, in this paper, we characterize Green's relations on it: that is, we study equivalence relations which determine when principal left (or right, or 2-sided) ideals in P(X, Y, θ) are equal. This solves a problem raised by Magill and Subbiah in 1975. We also discuss the same idea for important subsemigroups of P(X, Y, θ) and characterize when these semigroups satisfy certain regularity conditions. © 2013 World Scientific Publishing Company.

U2 - 10.1142/S179355711350006X

DO - 10.1142/S179355711350006X

M3 - Article

VL - 6

SP - 1350006-1 - 1350006-11

JO - Asian-European Journal of Mathematics

JF - Asian-European Journal of Mathematics

SN - 1793-5571

IS - 1

ER -