Semidistributivity and Whitman Property in implication zroupoids

In 2012, the second author introduced, and initiated the investigations into, the variety 𝓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. 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: (x → y) → z ≈ [(z′ → x) → (y → z)′]′, where x′ := x → 0, and 0″ ≈ 0. Let 𝓘 denote the variety of implication zroupoids and A ∈ 𝓘. For x, y ∈ A, let x ∧ y := (x → y′)′ and x ∨ y := (x′ ∧ y′)′. In an earlier paper, we had proved that if A ∈ 𝓘, then the algebra A mj = 〈A, ∨, ∧〉 is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every A ∈ 𝓘, the bisemigroup A mj is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety 𝓜𝓔𝓙 of 𝓘, defined by the identity: x ∧ y ≈ x ∨ y, satisfies the Whitman Property. We conclude the paper with two open problems.

Logics of Involutive Stone Algebras

An involutive Stone algebra (IS-algebra) is a structure that is simultaneously a De Morgan algebra and a Stone algebra (i.e. a pseudo-complemented distributive lattice satisfying the well-known Stone identity, ∼ x ∨ ∼ ∼ x ≈ 1). IS-algebras have been studied algebraically and topologically since the 1980’s, but a corresponding logic (here denoted IS ≤ ) has been introduced only very recently. The logic IS ≤ is the departing point for the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that IS ≤ is a conservative expansion of the Belnap-Dunn four-valued logic (i.e. the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic so as to obtain an extension of IS ≤ . We show that every logic thus defined can be axiomatized by adding a fixed finite set of rule schemata to the corresponding super-Belnap base logic. We also consider a few sample extensions of IS ≤ that cannot be obtained in the above- described way, but can nevertheless be axiomatized finitely by other methods. Most of our axiomatization results are obtained in two steps: through a multiple-conclusion calculus first, which we then reduce to a traditional one. The multiple-conclusion axiomatizations introduced in this process, being analytic, are of independent interest from a proof-theoretic standpoint. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of IS ≤ . Indeed, as in the super-Belnap case, we establish that the finitary extensions of IS ≤ are already uncountably many.

Notes on the Logic of Perfect Paradefinite Algebras

This work introduces the variety of perfect paradefinite algebras (PPalgebras), consisting of De Morgan algebras enriched with a perfect operator o, which turns out to be equivalent to the variety of involutive Stone algebras (IS-algebras). The corresponding order-preserving logic PP≤ is a Logic of Formal Inconsistency, a Logic of Formal Undeterminedness, a C-system and a D-system, some of these features being evident in the proposed axiomatization of PP-algebras. After proving the mentioned algebraic equivalence, we show how to axiomatize, by means of Hilbert-style calculi, certain extensions of De Morgan algebras with a perfect operator and, in particular, the logic PP≤.

Tense De Morgan S4-algebras

In [Tense operators on De Morgan algebras, Log. J. IGPL 22(2) (2014) 255–267], Figallo and Pelaitay introduced the notion of tense operators on De Morgan algebras. Also, other notions of tense operators on De Morgan algebras were given by Chajda and Paseka in [De Morgan algebras with tense operators, J. Mult.-Valued Logic Soft Comput. 1 (2017) 29–45; The Poset-based logics for the De Morgan negation and set representation of partial dynamic De Morgan algebras, J. Mult.-Valued Logic Soft Comput. 31(3) (2018) 213–237; Set representation of partial dynamic De Morgan algebras, in 2016 IEEE 46th Int. Symp. Multiple-Valued Logic (IEEE Computer Society, 2016), pp. 119–124]. In this paper, we introduce a new notion of tense operators on De Morgan algebras and define the class of tense De Morgan [Formula: see text]-algebras. The main purpose of this paper is to give a discrete duality for tense De Morgan [Formula: see text]-algebras. To do this, we will extend the discrete duality given in [W. Dzik, E. Orłowska and C. van Alten, Relational Representation Theorems for Lattices with Negations: A Survey, Lecture Notes in Computer Science (2006), pp. 245–266], for De Morgan algebras.

Some De Morgan algebras associated with a double fuzzy topological space

This paper identifies a De Morgan algebra associated with the families of (r, s)-regular fuzzy open sets and (r, s)-regular fuzzy closed sets in a double fuzzy topological space. The situation under which this De Morgan algebra becomes a Boolean algebra is characterized. Certain other properties of this algebra are also investigated.

Discrete duality for De Morgan Algebras with operators

A discrete duality is a relationship between classes of algebras and classes of relational systems (frames). In this paper, discrete dualities are presented for De Morgan algebras with various kind of unary operators. To do this, we will extend the discrete duality given in [W. Dzik, E. Orłowska and C. van Alten, Relational representation theorems for general lattices with negations, in Relations and Kleene Algebra in Computer Science, Lecture Notes in Computer Science, Vol. 4136 (Springer, Berlin, 2006), pp. 162–176], for De Morgan algebras.

Free Modal Pseudocomplemented De Morgan Algebras

Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ(∼x)* = (∼(xΛ(∼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18].