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In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák's theorem, in the sense that in the variety ℝ D ℙ ℂ ℍ of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák's formula. As applications of this result together with the above-mentioned Katriňák's theorem, we show that the varieties ℝ D B L ℙ , ℝ D ℙ ℂ ℍ , ℝ ℙ ℂ ℍ d and ℝ D B L ℍ of regular double p-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝ D M ℙ , ℝ D M ℍ , ℝ D M D B L ℍ , ℝ D M D B L ℙ of regular De Morgan p-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2 ℵ 0 . We then define new logics, ℛ D P C ℋ , ℛ P C ℋ d and ℛ D ℳ ℋ , and show that they are algebraizable with ℝ D ℙ ℂ ℍ , ℝ ℙ ℂ ℍ d and ℝ D M ℍ , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2 ℵ 0 . [ABSTRACT FROM AUTHOR]

In this paper, we provide a new characterization of ℒ -fuzzy ideals of residuated lattices, which allows us to describe ℒ -fuzzy ideals generated by ℒ -fuzzy sets. Thanks to the latter, we endow the lattice of ℒ -fuzzy ideals of a residuated lattice with suitable operations. Moreover, we introduce the notion of ℒ -fuzzy annihilator of an ℒ -fuzzy subset of a residuated lattice with respect to an ℒ -fuzzy ideal and investigate some of its properties. To this extent, we show that the set of all ℒ -fuzzy ideals of a residuated lattice is a complete Heyting algebra. Furthermore, we define some types of ℒ -fuzzy ideals of residuated lattices, namely stable ℒ -fuzzy ideals relative to an ℒ -fuzzy set, and involutory ℒ -fuzzy ideals relative to an ℒ -fuzzy ideal. Finally, we prove that the set of all stable ℒ -fuzzy ideals relative to an ℒ -fuzzy set is also a complete Heyting algebra, and that the set of involutory ℒ -fuzzy ideals relative to an ℒ -fuzzy ideal is a complete Boolean algebra. [ABSTRACT FROM AUTHOR]

In the paper, we introduce 𝔏-fuzzy state filters in state residuated lattices and investigate their related properties, where 𝔏 is a complete Heyting algebra. Moreover, we study the 𝔏-fuzzy state co-annihilator of an 𝔏-fuzzy set with respect to an 𝔏-fuzzy state filter. Finally, using the 𝔏-fuzzy state co-annihilator, we investigate lattice structures of the set of some types of 𝔏-fuzzy state filters in state residuated lattices. In particular, we prove that: (1) the set FSF[L] of all 𝔏-fuzzy state filters is a complete Heyting algebra; (2) the set SνFSF[L] of all stable state filters relative to an 𝔏-fuzzy set ν is also a complete Heyting algebra; (3) the set IμFSF[L] of all involutory 𝔏-fuzzy state filters relative to an 𝔏-fuzzy state filter μ is a complete Boolean algebra. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP1, LV(RDP1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP1 and conclude that LV(RDP1) is countably infinite. An equational basis for each proper subvariety of RDP1 is given. To study the subvarieties RDPn with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2ℵ0. We also use Priestley duality to prove that RDP and each of its subvarieties RDPn are generated by their finite members. [ABSTRACT FROM AUTHOR]

The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH1 of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity and present (equational) axiomatizations for several subvarieties of RDMSH1. Secondly, using our earlier results published in 2014, we give a concrete description of the lattice of subvarieties of the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras that contains RDMSH1. Furthermore, we prove that every subvariety of RDQDStSH1, and hence of RDMSH1, has Amalgamation Property. The note concludes with some open problems for further investigation. [ABSTRACT FROM AUTHOR]

The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH1 of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity and present (equational) axiomatizations for several subvarieties of RDMSH1. Secondly, using our earlier results published in 2014, we give a concrete description of the lattice of subvarieties of the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi- Heyting algebras that contains RDMSH1. Furthermore, we prove that every subvariety of RDQDStSH1, and hence of RDMSH1, has Amalgamation Property. The note concludes with some open problems for further investigation. [ABSTRACT FROM AUTHOR]