Model-completion of varieties of co-Heyting algebras

It is known that exactly eight varieties of Heyting algebras have a model-completion, but no concrete axiomatisation of these model-completions were known by now except for the trivial variety (reduced to the one-point algebra) and the variety of Boolean algebras. For each of the six remaining varieties we introduce two axioms and show that 1) these axioms are satisfied by all the algebras in the model-completion, and 2) all the algebras in this variety satisfying these two axioms have a certain embedding property. For four of these six varieties (those which are locally finite) this actually provides a new proof of the existence of a model-completion, this time with an explicit and finite axiomatisation.Comment: 28 page

Expanding FLew with a Boolean connective

We expand FLew with a unary connective whose algebraic counterpart is the operation that gives the greatest complemented element below a given argument. We prove that the expanded logic is conservative and has the Finite Model Property. We also prove that the corresponding expansion of the class of residuated lattices is an equational class.Comment: 15 pages, 4 figures in Soft Computing, published online 23 July 201

A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions

Semi-Heyting algebras were introduced by the second-named author during 1983-85 as an abstraction of Heyting algebras. The first results on these algebras, however, were published only in 2008 (see [San08]). Three years later, in [San11], he initiated the investigations into the variety DHMSH of dually hemimorphic semi-Heyting algebras obtained by expanding semi-Heyting algebras with a dually hemimorphic operation. His investigations were continued in a series of papers thereafter. He also had raised the problem of finding logics corresponding to subvarieties of DHMSH, such as the variety DMSH of De Morgan semi-Heyting algebras, and DPCSH of dually pseudocomplemented semi-Heyting algebras, as well as logics to 2, 3, and 4-valued DHMSH-matrices. In this paper, we first present a Hilbert-style axiomatization of a new implicative logic called--Dually hemimorphic semi-Heyting logic, (DHMSH, for short)-- as an expansion of semi-intuitionistic logic by the dual hemimorphism as the negation and prove that it is complete with respect to the variety DHMSH of dually hemimorphic semi-Heyting algebras as its equivalent algebraic semantics (in the sense of Abstract Algebraic Logic). Secondly, we characterize the (axiomatic) extensions of DHMSH in which the Deduction Theorem holds. Thirdly, we present several logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH, thus solving the problem mentioned earlier. We also provide new axiomatizations for Moisil's logic and the 3-valued Lukasiewicz logic.Comment: 66 pages, 3 figure