Rough relation algebras1,2

Rough relation algebras were introduced by S. Comer as a generalisation of algebras of Pawlak's rough sets and Tarski's relation algebras. In this paper, some algebraic and arithmetical properties of rough relation algebras are studied and the representable rough relation algebras are characterised. 1. Definitions and preliminary results We assume that the reader is familiar with the basic facts of relation algebras as presented e.g. in [4] or [8]. For Stone algebras the reader is invited to consult [2]. Rough sets and rough relations were introduced by Z. Pawlak ([10], [11]) arising from his work on approximation spaces and information systems, and presented as an alternative to fuzzy sets and tolerance theory. Rough relations can be thought of as an approximation of relations via an equivalence relation on the base set when there is only incomplete information. A description of the construction is given below. A lattice theoretic approach to rough sets was taken by Iwinski [6], and in [12] it was shown that the collection of rough sets of an approximation space forms a Stone algebra with appropriately defined operations. Furthermore, it turns out that the set of all rough relations on a set can be made into a regular double Stone algebra in a natural way [3]. A double Stone algebra (DSA) is an algebra of type <2,2,1,1,0,0> such that 1. is a bounded distributive lattice, 2. x* is the pseudocomplement of x, i.e. y ≤ x* ⇔ y⋅x = 0, 3. x+ is the dual pseudocomplement of x, i.e. y ≥ x+ ⇔ y + x = 1 4. x* + x** = 1, x+ ⋅ x++ = 0. Conditions 2. and 3. are equivalent to the equations x⋅(x⋅y)* = x⋅y*, x + (x + y)+ = x + x+ x⋅0* = x, x + 1+ = x 0** = 0, 1++ = 1 1The paper was inspired by Steve Comer's work [C] and I should like to thank him for fruitful discussions. 2AMS Classification 03G15 so that DSA is an equational class. L is called regular, if it additionally satisfies the equation x ⋅ x+ ≤ x + x*. This is equivalent to (x+ = y+ ∧ x* = y*) ⇒ x = y. The centre B(L) = {x*: x ∈ L} of L is a subalgebra of L and a Boolean algebra, in which * and + coincide with the Boolean complement. The dense set {x ∈ L: x* = 0} of L is denoted by D(L), or simply D, if L is understood. For any M ⊆ L, M+ is the set {x+: x ∈ M}. The first lemma collects a few well known arithmetical properties of double Stone algebras [2] which will be useful in the sequel: