Mitsch’s order and inclusion for binary relations and partitions

Abstract

Abstract. Mitsch's natural partial order on the semigroup of binary relations is here characterised by equations in the theory of relation algebras. The natural partial order has a complex relationship with the compatible partial order of inclusion, which is explored by means of a sublattice of the lattice of preorders on the semigroup. The corresponding sublattice for the partition monoid is also described. Natural partial ordersIn [10], Heinz Mitsch formulated a characterisation of the natural partial order ≤ on the full transformation semigroup T X which did not use inverses or idempotents, and went on to define the natural partial order ≤ on any semigroup S by(The discovery was also made independently by P. M. Higgins, but remained unpublished.) Observe that a = ya follows. Mitsch's natural partial order has now been characterised, and its properties investigated, for several concrete classes of non-regular semigroups-in [8,12] for some semigroups of (partial) transformations, and by Namnak and Preechasilp [11] for the semigroup B X of all binary relations on the set X. The partial order of inclusion which is carried by B X may also be thought of as 'natural', and it is the broad purpose of this note to discuss the relationship between these two partial orders on B X . Moreover, the same questions are addressed for the partition monoid P X , which also carries two 'natural' partial orders. So we shall use a slightly different nomenclature here for the sake of clarity, mostly referring to partial orders as just orders, and the natural partial order as Mitsch's order. We begin by collecting some information about B X . Binary relationsThe notation used here for binary relations follows that found in, for example, Clifford and Preston [1], with the addition of complementation of relations defined by x α c y ⇐⇒ (x, y) ∈ α for x, y ∈ X. Note that the symbol • for composition will be suppressed, except for the composites of order relations on B X . We will make use of the identity relation on X, ι = {(x, x) : x ∈ X} , and the universal relation ω = X × X.The following logical equivalence will also be required; it is the 'Theorem K' of De Morgan [2, p. xxx ] (see also e.g. [6]).

Ehresmann theory and partition monoids

“…The role played in Section 3 by surjections between subsets of X is here played by injections between quotients of X. Other parallels between B X and P X have been explored in [29,30].…”

Ehresmann theory and partition monoids

“…The role played in Section 3 by surjections between subsets of X is here played by injections between quotients of X. Other parallels between B X and P X have been explored in [24,25].…”