Congruences on Dynamic Algebras

Tense operators for MV-algebras were introduced by Diaconescu and Georgescu. Based on their denition Chajda and Kolařik presented the denition of tense operators for lattice effect algebras. Chajda and Paseka tackled the problem of axiomatizing tense operators on an effect algebra by introducing the notion of a partial dynamic effect algebra. They also gave representation theorems for dynamic effect algebras. We continue to extend their work for partial S-dynamic effect algebras i.e. in the case when tense operators satisfy required conditions also for the dual effect algebraic operation . We show that whenever tense operators are total our stronger notion coincides with their denition. We give also a representation theorem for partial S-dynamic effect algebras and its version for strict dynamic effect algebras.

The present paper is an attempt to suggest and scrutinize tense operators in the dynamic logic $textbf{B}$ which is regarded as a set of propositions about the general fuzzy automaton $ tilde{F} $, in which its underlying structure has been a bounded poset. Here, the operators $ T_{delta}, P_{delta}, H_{delta}$ and $ F_{delta} $ are proposed regardless of what propositional connectives the logic comprises. For this purpose, the axiomatization of universal quantifiers is applied as a starting point and these axioms are modified. In this study, firstly, we demonstrate that the operators can be identified as modal operators and the pairs $ (T_{delta},P_{delta}) $ are examined as the so-called dynamic pairs. In addition, constructions of these operators are attained in the corresponding algebra and in the following a transition frame is suggested. Besides, the problem of finding a transition frame is solved in the case when the tense operators are given. Specifically, this study shows that the tense algebra $ textbf{B} $ is representable in its Dedekind-MacNeille completion. Representation theorems for dynamic and tense algebra are explicated in details in the related given theorems.