Partial Orders on Partial Baer–levi Semigroups

Abstract

Marques-Smith and Sullivan ['Partial orders on transformation semigroups ', Monatsh. Math. 140 (2003), 103-118] studied various properties of two partial orders on P(X ), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X . One partial order was the 'containment order': namely, if α, β ∈ P(X ) then α ⊆ β means xα = xβ for all x ∈ dom α, the domain of α. The other order was the so-called 'natural order' defined by Mitsch ['A natural partial order for semigroups', Proc. Amer. Math. Soc. 97(3) (1986), 384-388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I (X ) and the partial Baer-Levi semigroup P S(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.2000 Mathematics subject classification: primary 20M20; secondary 04A05, 06A06.

Injective Partial Transformations With Infinite Defects

“…However, R(p) equals the semigroup generated by all of the nilpotents in I(X). Also, as in [11] Remark (with a small correction), if p = q, then P S(p) is the union of R(p) and the set of elements in P S(p) which are maximal under ≤, and the latter set forms a semigroup.…”

“…If g(γ) < q, then [11,Theorem 4.3] implies that γ is maximal under ≤ and so γ = α = β, contradicting the supposition. Hence g(γ) ≥ q.…”

“…In effect, by [11,Theorem 4.3], the next result determines when two elements of P S(q), which are maximal under ≤, possess a meet under ≤. …”

“…In [11] the authors observed that ≤ = ⊆ and Ω = Ω ′ on I(X), but ≤ does not equal Ω on I(X). In Section 6, we define a new partial order on any inverse semigroup, show that it equals Ω on I(X) and discuss some of its algebraic properties.…”

“…In Section 6, we define a new partial order on any inverse semigroup, show that it equals Ω on I(X) and discuss some of its algebraic properties. On the other hand, it was shown in [11] that, when restricted to P S(q), ≤ is properly contained in ⊆, and ⊆ is properly contained in Ω. In Sections 4 and 5, we characterize the meet and join for elements of R(q) and P S(q) under ≤ and ⊆.…”