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 -