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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 2012, the second author introduced and studied in Sankappanavar (Sci Math Jpn 75(1):21-50, 2012) the variety $${\mathcal {I}}$$ of algebras, called implication zroupoids, that generalize De Morgan algebras. An algebra $${\mathbf {A}} = \langle A, \rightarrow , 0 \rangle $$ , where $$\rightarrow $$ is binary and 0 is a constant, is called an implication zroupoid ( $${\mathcal {I}}$$ -zroupoid, for short) if $${\mathbf {A}}$$ satisfies: $$(x \rightarrow y) \rightarrow z \approx [(z' \rightarrow x) \rightarrow (y \rightarrow z)']'$$ and $$ 0'' \approx 0$$ , where $$x' : = x \rightarrow 0$$ . The present authors devoted the papers, Cornejo and Sankappanavar (Alegbra Univers, 2016a; Stud Log 104(3):417-453, 2016b. doi:; and Soft Comput: 20:3139-3151, 2016c. doi:), to the investigation of the structure of the lattice of subvarieties of $${\mathcal {I}}$$ , and to making further contributions to the theory of implication zroupoids. This paper investigates the structure of the derived algebras $$\mathbf {A^{m}} := \langle A, \wedge , 0 \rangle $$ and $$\mathbf {A^{mj}} :=\langle A, \wedge , \vee , 0 \rangle $$ of $${\mathbf {A}} \in {\mathcal {I}}$$ , where $$x \wedge y := (x \rightarrow y')'$$ and $$x \vee y := (x' \wedge y')'$$ , as well as the lattice of subvarieties of $${\mathcal {I}}$$ . The varieties $${\mathcal {I}}_{2,0}$$ , $${{\mathcal {R}}}{{\mathcal {D}}}$$ , $$\mathcal {SRD}$$ , $${\mathcal {C}}$$ , $${{\mathcal {C}}}{{\mathcal {P}}}$$ , $${\mathcal {A}}$$ , $${{\mathcal {M}}}{{\mathcal {C}}}$$ , and $$\mathcal {CLD}$$ are defined relative to $${\mathcal {I}}$$ , respectively, by: (I $$_{2,0}$$ ) $$x'' \approx x$$ , (RD) $$(x \rightarrow y) \rightarrow z \approx (x \rightarrow z) \rightarrow (y \rightarrow z)$$ , (SRD) $$(x \rightarrow y) \rightarrow z \approx (z \rightarrow x) \rightarrow (y \rightarrow z)$$ , (C) $$ x \rightarrow y \approx y \rightarrow x$$ , (CP) $$ x \rightarrow y' \approx y \rightarrow x'$$ , (A) $$(x \rightarrow y) \rightarrow z \approx x \rightarrow (y \rightarrow z)$$ , (MC) $$x \wedge y \approx y \wedge x$$ , (CLD) $$x \rightarrow (y \rightarrow z) \approx (x \rightarrow z) \rightarrow (y \rightarrow x)$$ . The purpose of this paper is two-fold. Firstly, we show that, for each $${\mathbf {A}} \in {\mathcal {I}}$$ , $${\mathbf {A}}^{\mathbf {m}}$$ is a semigroup. From this result, we deduce that, for $${\mathbf {A}} \in {\mathcal {I}}_{2,0} \cap {{\mathcal {M}}}{{\mathcal {C}}}$$ , the derived algebra $$\mathbf {A^{mj}}$$ is a distributive bisemilattice and is also a Birkhoff system. Secondly, we show that $$\mathcal {CLD} \subset \mathcal {SRD} \subset {{\mathcal {R}}}{{\mathcal {D}}}$$ and $${\mathcal {C}} \subset \ {{\mathcal {C}}}{{\mathcal {P}}} \cap {\mathcal {A}} \cap {{\mathcal {M}}}{{\mathcal {C}}} \cap \mathcal {CLD}$$ , both of which are much stronger results than were announced in Sankappanavar (Sci Math Jpn 75(1):21-50, 2012). [ABSTRACT FROM AUTHOR]

In a paper published in 2012, the second author extended the well-known fact that Boolean algebras can be defined using only implication and a constant, to De Morgan algebras-this result led him to introduce, and investigate (in the same paper), the variety $${\mathcal{I}}$$ of algebras, there called implication zroupoids ( I-zroupoids) and here called implicator groupoids ( $${\mathcal{I}}$$ -groupoids), that generalize De Morgan algebras. The present paper is a continuation of the paper mentioned above and is devoted to investigating the structure of the lattice of subvarieties of $${\mathcal{I}}$$ , and also to making further contributions to the theory of implicator groupoids. Several new subvarieties of $${\mathcal{I}}$$ are introduced and their relationship with each other, and with the subvarieties of $${\mathcal{I}}$$ which were already investigated in the paper mentioned above, are explored. [ABSTRACT FROM AUTHOR]

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In fact, in 1934, Bernstein (Trans Am Math Soc 36:876-884, 1934) gave a system of axioms for Boolean algebras in terms of implication only. Though his original axioms were not equational, a quick look at his axioms would reveal that if one adds a constant, then it is not hard to translate his system of axioms into an equational one. Recently, in 2012, the second author of this paper extended this modified Bernstein's theorem to De Morgan algebras (see Sankappanavar, Sci Math Jpn 75(1):21-50, 2012). Indeed, it is shown in Sankappanavar (Sci Math Jpn 75(1):21-50, 2012) that the varieties of De Morgan algebras, Kleene algebras, and Boolean algebras are term-equivalent, respectively, to the varieties, $$\mathbf {DM}$$ , $$\mathbf {KL}$$ , and $$\mathbf {BA}$$ whose defining axioms use only the implication $$\rightarrow $$ and the constant 0. The fact that the identity, herein called (I), occurs as one of the two axioms in the definition of each of the varieties $$\mathbf {DM}$$ , $$\mathbf {KL}$$ and $$\mathbf {BA}$$ motivated the second author of this paper to introduce, and investigate, the variety $$\mathbf {I}$$ of implication zroupoids, generalizing De Morgan algebras. These investigations are continued by the authors of the present paper in Cornejo and Sankappanavar (Implication zroupoids I, 2015), wherein several new subvarieties of $$\mathbf {I}$$ are introduced and their relationships with each other and with the varieties studied in Sankappanavar (Sci Math Jpn 75(1):21-50, 2012) are explored. The present paper is a continuation of Sankappanavar (Sci Math Jpn 75(1):21-50, 2012) and Cornejo and Sankappanavar (Implication zroupoids I, 2015). The main purpose of this paper is to determine the simple algebras in $$\mathbf {I}$$ . It is shown that there are exactly five (nontrivial) simple algebras in $$\mathbf {I}$$ . From this description we deduce that the semisimple subvarieties of $$\mathbf {I}$$ are precisely the subvarieties of the variety generated by these simple I-zroupoids and that they are locally finite. It also follows that the lattice of semisimple subvarieties of $$\mathbf {I}$$ is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain. [ABSTRACT FROM AUTHOR]