ABSTRACT In this paper, we present the corresponding algebraic structure for ‐modal ‐logic (abbreviated as ) proposed by Eslami et al. [ Journal of Intelligent and Fuzzy Systems ] 29 (2015): 337–388, which we call a strong ‐modal ‐algebra. We demonstrate that the equivalent algebraic semantics for is the variety of strong ‐modal ‐algebras, and is algebraizable in the sense of Blok and Pigozzi. We prove that is strongly sound and complete with respect to strong ‐modal ‐algebras. The local deduction theorem is also established for . Furthermore, we show that is strongly sound and complete with respect to linearly ordered strong ‐modal ‐algebras. We show that is a conservative expansion of Hájek's basic logic. Finally, using algebraizability, we analyze congruences in strong ‐modal ‐algebras.
Algebras of disjoint alternatives (ADAs), introduced by Manes, are extensions of McCarthy algebras by a unary operator acting as an oracle for the halting problem, and provide a semantics for the equational theory of the if-then-else construct.We study the variety of ADAs in the signature without the constant of Manes' formulation, and prove a representation theorem: every ADA has an essentially unique representation as an algebra of pairs , where is a Boolean algebra and a distinguished filter of . This refines Manes' construction, which captures the ADAs possessing a fixed point of the negation, i.e. the case .Since regular double Stone algebras admit a similar representation due to Katri & ncaron;& aacute;k, we derive that the two varieties are term-equivalent, and hence so are ADAs, three-valued & Lstrok;ukasiewicz algebras, and three-valued MV-algebras. As consequences, we obtain a categorical equivalence between ADAs and Boolean algebras with a distinguished filter, extending Manes' equivalence, and the amalgamation property for the variety of ADAs.The representation accommodates, in a single framework, the connectives of Kleene's strong, McCarthy's sequential, and weak Kleene logics, modeling three distinct forms of error propagation in computation. Finally, we outline exploratory connections with rough sets and assume-guarantee contracts.
ABSTRACT In this paper, we propose the strong* proximity lattices, and prove that the categories and are categorically equivalent, building on the framework established by Achim Jung and Philipp Sünderhauf. denotes the category of all QFS‐domains and continuous functions. denotes the category of the strong* proximity lattices with the approximable relations. This answers the question posed by Achim Jung at the conference ISDT'13.
We introduce an -ary operation on the class of the ordinals, which is strictly monotone in all significant cases. We provide order-theoretical characterizations as the rank of a sequence in a well-founded order and as a mixed sum of the ordinals in the sequence.
In this study, we extend the additively generated triangular norms from the framework of the unit interval to that of partially ordered sets. We present several conditions under which this formula yields a t-norm, where are partially ordered sets, and are monotone functions and is a t-norm/t-conorm on . The partially ordered semigroups induced by are order-preserving/order-reversing homomorphic to semigroup deformations of the semigroup induced by .
