The logic of tied implications, part 1: Properties, applications and representation

Abstract

A conjunction T ties an implication operator A if the identity A(a, A(b, z)) = A(T (a, b), z) holds [We study the class of tied adjointness algebras (which are five-connective algebras on two partially ordered sets), in which the implications are tied by triangular norms. This class contains, besides residuated implications, several other implications employed in fuzzy logic. Nevertheless, we show that the algebraic inequalities of residuated algebras remain true for our tied implications, but in forms that distribute roles over the five connectives of the algebra. We apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules. We prove its equivalence to a scheme with one compound rule, when both schemata are interpreted by the compositional rule of inference, and all connectives are taken from one tied adjointness algebra. Then we quote another application of this rich theory, a notion of many-valued rough sets, which exhibit the basic mathematical behaviour of the rough sets of Pawlak. A comparator H is said to be prelinear if it satisfies H (y, z) ∨ H (z, y) = 1 for all y, z (Hájek). We introduce prelinear tied adjointness algebras, in which two comparators are prelinear. We provide a representation of those algebras, as subdirect products of tied adjointness chains, on the lines of Hájek's representation of BL-algebras. But our representations are more economical, because we employ minimal prime filters (on residuated lattices) only; rather than all prime filters.

The logic of tied implications, part 2: Syntax

“…The chain completeness of L-AdjTPC, with respect to all tied adjointness chains, is derived also through the representation theorem established in Part 1 [15].…”

“…We devote this subsection to a brief summary of the concepts and results of the first part [15] of this work. Throughout, (P , P ) and (L, L ) denote partially ordered sets (posets).…”

“…We need the following identity satisfied [15] by all comparators H in all adjointness algebras: for any subset {z j } j ∈ of L that has a supremum in L, and any subset {w k } k∈ of L that has an infimum in L:…”

“…Definition 2.1.3 (Morsi [14,15]). A tied adjointness algebra is a mathematical system (L, L , P , P , 1, A, K, H, T , I ) in which (L, L , P , P , 1, A, K, H ) is an adjointness algebra, (P , P , 1, T , I ) is a residuated algebra, and T ties A; that is,…”

“…We continue the study of tied adjointness algebras, begun in [1,14] and Part 1 [15]. The implications A of those algebras are tied by commutative integral ordered monoid operations T in the following manner:…”