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We present a category equivalent to that of semi-Nelson algebras. The objects in this category are pairs consisting of a semi-Heyting algebra and one of its filters. The filters must contain all the dense elements of the semi-Heyting algebra and satisfy an additional technical condition. We also show that the category of dually hemimorphic semi-Nelson algebras is equivalent to that of dually hemimorphic semi-Heyting algebras. [ABSTRACT FROM AUTHOR]

An algebra A = ⟨ A , → , 0 ⟩ , where → is binary and 0 is a constant, is called an implication zroupoid (I -zroupoid, for short) if A satisfies the identities: (x → y) → z ≈ ((z ′ → x) → (y → z) ′) ′ and 0 ′ ′ ≈ 0 , where x ′ : = x → 0 . An implication zroupoid is symmetric if it satisfies: x ′ ′ ≈ x and (x → y ′) ′ ≈ (y → x ′) ′ . The variety of symmetric I -zroupoids is denoted by S . We began a systematic analysis of weak associative laws (or identities) of length ≤ 4 in Cornejo and Sankappanavar (Soft Comput 22(13):4319–4333, 2018a. 10.1007/s00500-017-2869-z), by examining the identities of Bol–Moufang type, in the context of the variety S . In this paper, we complete the analysis by investigating the rest of the weak associative laws of length ≤ 4 relative to S . We show that, of the (possible) 155 subvarieties of S defined by the weak associative laws of length ≤ 4 , there are exactly 6 distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of S defined by weak associative laws of length ≤ 4 . [ABSTRACT FROM AUTHOR]

An algebra A=⟨A,→,0⟩, where → is binary and 0 is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies the identities: (I): (x→y)→z≈((z′→x)→(y→z)′)′, and (I0): 0′′≈0, where x′:=x→0. An implication zroupoid is symmetric if it satisfies the identities: x′′≈x and (x→y′)′≈(y→x′)′. An identity is of Bol-Moufang type if it contains only one binary operation symbol, one of its three variables occurs twice on each side, each of the other two variables occurs once on each side, and the variables occur in the same (alphabetical) order on both sides of the identity. In this paper, we will present a systematic analysis of all 60 identities of Bol-Moufang type in the variety S of symmetric I-zroupoids. We show that 47 of the subvarieties of S, defined by the identities of Bol-Moufang type, are equal to the variety SL of ∨-semilattices with the least element 0 and one of others is equal to S. Of the remaining 12, there are only three distinct ones. We also give an explicit description of the poset of the (distinct) subvarieties of S of Bol-Moufang type. [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]

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]