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ntly, Gautama algebras were defined and investigated as a common generalization of the variety RDBLSt of regular double Stone algebras and the variety RKLSt of regular Kleene Stone algebras, both of which are, in turn, generalizations of Boolean algebras. Those algebras were named in honor and memory of the two founders of Indian Logic{Akshapada Gautama and Medhatithi Gautama. The purpose of this paper is to define and investigate a generalization of Gautama algebras, called \Almost Gautama algebras (AG, for short)." More precisely, we give an explicit description of subdirectly irreducible Almost Gautama algebras. As consequences, explicit description of the lattice of subvarieties of AG and the equational bases for all its subvarieties are given. It is also shown that the variety AG is a discriminator variety. Next, we consider logicizing AG; but the variety AG lacks an implication operation. We, therefore, introduce another variety of algebras called \Almost Gautama Heyting algebras" (AGH, for short) and show that the variety AGH is term-equivalent to that of AG. Next, a propositional logic, called AG (or AGH), is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety AG, via AGH; as its equivalent algebraic semantics (up to term equivalence). All axiomatic extensions of the logic AG, corresponding to all the subvarieties of AG are given. They include the axiomatic extensions RDBLSt, RKLSt and G of the logic AG corresponding to the varieties RDBLSt, RKLSt, and G (of Gautama algebras), respectively. It is also deduced that none of the axiomatic extensions of AG has the Disjunction Property. Finally, We revisit the classical logic with strong negation CN and classical Nelson algebras CN introduced by Vakarelov in 1977 and improve his results by showing that CN is algebraizable with CN as its algebraic semantics and that the logics RKLSt, RKLStH, 3-valued Lukasivicz logic and the classical logic with strong negation are all equivalent [ABSTRACT FROM AUTHOR]

In this paper we study a subdirect irreducibility of algebras with one operator and the main near-unanimity operation defined by the specific way. It is shown that congruence lattices of algebras of given class are atomic. In given class, the full description of atoms in congruence lattices of algebras, of subdirectly irreducible algebras, and of algebras with atomistic congruence lattices is obtained. [ABSTRACT FROM AUTHOR]

In this work, we initiate an investigation of the class B T k m of monadic Boolean algebras endowed with a monadic automorphism of period k. These algebras constitute a generalization of monadic symmetric Boolean algebras. We determine the congruences on these algebras and we characterize the subdirectly irreducible algebras. This last result allows us to prove that B T k m is a discriminator variety and as a consequence, the principal congruences are characterized. Finally, we explore, in the finite case, the relationship between this class and the class Df 2 of diagonal-free two-dimensional cylindric algebras. [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]

A De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices). In this paper we give a functional representation of the free n-generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on nfree generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of nvariables. [ABSTRACT FROM AUTHOR]

A Boolean quasilattice is an algebra with hyperidentities of the variety of Boolean algebras. In this paper, we give a functional representation of the free n-generated Boolean quasilattice with two binary, one unary and two nullary operations. Namely, we define the concept of super-Boolean function and prove that the free Boolean quasilattice with two binary, one unary and two nullary operations on n free generators is isomorphic to the Boolean quasilattice of super-Boolean functions of n variables. [ABSTRACT FROM AUTHOR]

The paper characterizes the class of subdirectly irreducible algebras satisfying hyperidentities of the variety of De Morgan algebras. Such algebras are called subdirectly irreducible De Morgan quasilattices. The suggested characterization is quite close to that of the classical case of subdirectly irreducible DeMorgan algebras. [ABSTRACT FROM AUTHOR]

We introduce the variety ℒ n m, m ≥ 1 and n ≥ 2, of m-generalized Łukasiewicz algebras of order n and characterize its subdirectly irreducible algebras. The variety ℒ n m is semisimple, locally finite and has equationally definable principal congruences. Furthermore, the variety ℒ n m contains the variety of Łukasiewicz algebras of order n. [ABSTRACT FROM AUTHOR]

We consider the variety O of Ockham algebras and its subvarieties of the form P m,n ( m > n ≥0), sometimes with an additional condition. We use Priestley duality and a remarkable theorem of Urquhart to develop a simple method for determining the equational bases of the subvarieties. The axioms that we obtain have the same canonical form and involve few variables. We illustrate our method by the detailed study of the variety MS 2 and some considerations about P 3,2. [ABSTRACT FROM AUTHOR]