In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculated coproduct and pushout in a special case. Also, we constructed a free EQ-algebra on a singleton.

In this paper, we introduce and study a corresponding logic to equality-algebras and obtain some basic properties of this logic. We prove the soundness and completeness of this logic based on equality-algebras and local deduction theorem. We show that this logic is regularly algebraizable with respect to the variety of equality∆-algebras but it is not Fregean. Then we introduce the concept of equality∆-algebras and investigate some related properties. Also, we study ∆-deductive systems of equality∆-algebras. In particular, we prove that every (...) prelinear equality ∆-algebra is a subdirect product of linearly ordered equality∆-algebras. Finally, we construct prelinear equality ∆ logic and prove the soundness and strong completeness of this logic respect to prelinear equality∆-algebras. (shrink)

In this paper, we will briefly look at the history of mathematical fuzzy logic in Czechoslovakia starting from the 1970s and extending until 2009. The role of P. Ha ́jek in the development of fuzzy logic is especially emphasized.