Mitschs Order and Inclusion for Binary Relations

Mitsch’s natural partial order on the semigroup of binary relations has a complex relationship with the compatible partial order of inclusion. This relationship is explored by means of a sublattice of the lattice of preorders on the semigroup. The natural partial order is also characterised by equations in the theory of relation algebras. 1. Natural partial orders Having formulated a characterisation of the natural partial order on the full transformation semigroup TX which did not require inverses or idempotents, Mitsch [6] 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. 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 [5, 8] for some semigroups of (partial) transformations, and by Namnak and Preechasilp [7] 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 : 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 …rst collect some information about BX :