Equational L-Consequence

Abstract

AbstractWhile the equational consequence of Chapter 7 is a direct application of the concept of matrix consequence to E-matrices, the concept of equational L-consequence is an extension of Birkhoff’s “equational logic”; namely, in contrast to the equational logic, we assume that the equational L-consequence can be defined for any nonempty set of E-matrices. Also in this chapter, we show the connection between the equational consequence and the equational L-consequence and illustrate the latter concept with two examples: the class of Boolean algebras with equality and the class of Heyting algebras with equality. In addition, we define the Lindenbaum-Tarski matrices for the equational L-consequence and demonstrate the usefulness of this concept.