Tense and dynamic algebras related to GFA

Document Type : Original Article

Authors

1 Department of Mathemathics, Shiraz Branch, Islamic Azad University, Shiraz, Iran

2 Department of Mathemathics, Graduate University of Advanced Technology, Kerman, Iran

3 Khatam Alanbia University of Technology, Behbahan, Iran

Abstract

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.

Keywords