Rough Sets and Algebras of Relations

A survey of results is presented on relationships between the algebraic systems derived from the approximation spaces induced by information systems and various classes of algebras of relations. Rough relation algebras are presented and it is shown that they form a discriminator variety. A characterisation of the class of representable rough relation algebras is given. The family of closure operators derived from an approximation space is abstractly characterised as certain type of Boolean algebra with operators. A representation theorem is given which says that every such an algebra is isomorphic with a similar algebra that is derived from an information system. 1 Notation and definitions The history and the impact of rough sets as a means of modelling incomplete information are considered elsewhere in this Volume, so we shall be content to just state the basic notions. Let U be a set and θ an equivalence relation on U . The pair 〈U, θ〉 will be called an approximation space. For A ⊆ U , Au = ⋃ {θx : x ∈ A} is the upper approximation of A, and A ⊆ U , Ad = ⋃ {θx : θx ⊆ A} is its lower approximation. A rough subset of U with respect to θ is a pair 〈Xd, Xu〉 with X ⊆ U . The collection of all rough subsets of U with respect to θ is denoted by Sbr(U) , and will sometimes be called a full algebra of rough sets. If θ is understood, we shall just write Sbr(U) 2 Rough Sets and Regular Double Stone Algebras An algebraic approach to rough sets was first proposed in (Iw1). Iwinski’s aim – later extended by (PP1) – was to endow the rough subsets ofU with a natural algebraic structure. It turns out that regular double Stone algebras are the proper setting. A double Stone algebra (DSA) 〈L,+, ·, ∗, +, 0, 1〉 is an algebra of type 〈2, 2, 1, 1, 0, 0〉 such that 1. 〈L,+, ·, 0, 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+ y x · 0∗ = x, x + 1+ = x 0∗∗ = 0, 1++ = 1 so that DSA is an equational class. L is called regular, if it additionally satisfies the equation x ·x+ ≤ y + y∗. This is equivalent to x+ = y+ and x∗ = y∗ imply x = y. The center B(L) = {x∗ : x ∈ L} of L is a subalgebra of L and a Boolean algebra, in which ∗ and + coincide with the Boolean complement which we denote by −. An element of the centre of L will also be called a Boolean element. The dense set {x ∈ L : x∗ = 0} of L is denoted byD(L), or simply D, if L is understood. For any M ⊆ L, M is the set {x+ : x ∈M}. A construction of regular double Stone algebras which is important for our purposes is given by Lemma 2.1. (Ka1) Let 〈B,+, ·,−, 0, 1〉 be a Boolean algebra and F be a not necessarily proper filter on B. Set 〈B, F 〉 = {〈a, b〉 ∈ B × B : a ≤ b and − b+ a ∈ F} . Then, L = 〈B, F 〉 is a 0,1 – sublattice of B × B, and it becomes a regular double Stone algebra by setting 〈a, b〉∗ = 〈−b,−b〉, 〈a, b〉+ = 〈−a,−a〉 . Furthermore,B(L) ∼= B as Boolean algebras, and D(L) ∼= F as lattices. Note that B(L) = {〈a, a〉 : a ∈ B}, D(L) = {〈a, 1〉 : a ∈ F} . Conversely, if M is a regular double Stone algebra, B = B(M), F = D(M), then the mapping which assigns to each x ∈M the pair 〈x++, x∗∗〉 is an isomorphism between M and 〈B, F 〉. If F = B, then 〈B, F 〉 is also denoted by B[2]. In view of things to come it is useful to note that by Lemma 2.1 each element x of a regular double Stone algebra is uniquely described by the greatest Boolean element below x and the smallest Boolean element above x. Now, suppose that 〈U, θ〉 is an approximation space. We can view the classes of θ as atoms of a complete subalgebra of the Boolean algebra Sb(U). Conversely, any atomic complete subalgebra B of Sb(U) gives rise to an equivalence relation θ on U , and this correspondence is bijective. The elements of B are ∅ and the unions of classes of its associated equivalence relation. If {a} ∈ B, then, for every X ⊆ U we have