Mitschs Order and Inclusion for Binary Relations and Partitions

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. 1. Natural partial orders In [10], Heinz Mitsch formulated a characterisation of the natural partial order on the full transformation semigroup TX which did not use inverses or idempotents, and went on to de…ne the natural partial order on any semigroup S by (1) a b if a = b or there are x; y 2 S such that a = ax = bx = yb for a; b 2 S. (The discovery was also made independently by P. M. Higgins, but remained unpublished.)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 BX of all binary relations on the set X. The partial order of inclusion which is carried by BX 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 BX : Moreover, the same questions are addressed for the partition monoid PX ; which also carries two ‘natural’partial orders. So we shall use a slightly di¤erent 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 BX :