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The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order |$p$| and |$p$| -rings, |$p$| prime and lift this result to the axled Łukasiewicz–Moisil algebra |$L \cong B_s \times P$| and the ring |$\prod ^s F_2 \times \prod ^l F_p$|⁠ , where |$B_s$| is a Boolean algebra of order |$2^s$|⁠ , |$P$| a |$p$| -valued Post algebra of order |$p^l$| and |$F_p$| is the prime field of order |$p$|⁠. [ABSTRACT FROM AUTHOR]

The ADC method of proof search in propositional natural deduction proposed in 2000 proceeds bottom up first by Analysing the sequent into sub-goals by applying all possible introduction rules, and then by checking whether each of these sub-goals can be established using only elimination rules (Direct Chaining). This looks simpler than the worst case complexity (PSPACE) of the derivability problem for intuitionistic propositional logic.We investigate the complexity ofADC for various fragments. ADC derivability is polynomially decidable for the &,→-fragment by a generalization of a familiar direct chaining algorithm for Horn formulas. Adding ⋁ leads to a CoNP-complete fragment. A short counterexample in case the goal sequent is not ADC derivable is provided by witnessing all relevant disjunctions. Adding constant ⊥ preserves CoNP completeness. [ABSTRACT FROM AUTHOR]

We endow the partially ordered set of nonempty faces of the n-cube with a distinguished 0-dimensional face and two operations that naturally extend the Rota-Metropolis partial operations. Although the structures thus obtained turn out to be termequivalent to Post algebras of order 3, the inclusion order between faces coincides with the De Luca-Termini sharpening order, and yields a compact coNP-complete logic that tolerates a modicum of inconsistency. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the class of P2-Almost Distributive Lattices and Post Almost Distributive Lattices are equationally definable. [ABSTRACT FROM AUTHOR]

Using Vaggione's concept of central element in a double-pointed algebra, we introduce the notion of Boolean-like variety as a generalisation of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean-like varieties, which still retain many of the pleasing properties of Boolean algebras. We prove that a double-pointed variety is discriminator if and only if it is semi-Boolean-like, idempotent, and 0-regular. This theorem yields a new Maltsev-style characterisation of double-pointed discriminator varieties. [ABSTRACT FROM AUTHOR]

In the paper [5] all the solutions of Boolean inequation were described. In this paper we generalize the result from [5] to Post inequations i.e. to ineqations of the form f (X) ≠ 0, where f is a Post polynomial. The basic idea in this paper is: the inequation f (X) ≠ 0 means that there exists p such that f (X) = p and p ≠ 0. The solution of a Post inequation can be reduced to the solution of two Post equations. [ABSTRACT FROM AUTHOR]

A theorem due to Lowenheim provides the general reproductive solution of a Boolean equation whenever a particular solution is known. This theorem has been generalized either by passing from Boolean algebras to Post algebras and Stone algebras, or by using a parametric solution instead of a particular solution, or by combining the two directions. In the present paper we survey this research and generalize everything to Stone algebras. [ABSTRACT FROM AUTHOR]

There is a constructive method to define a structure of simple k-cyclic Post algebra of order p, L, on a given finite field F( p), and conversely. There exists an interpretation Φ of the variety $${\mathcal{V}(L_{p,k})}$$ generated by L into the variety $${\mathcal{V}(F(p^k))}$$ generated by F( p) and an interpretation Φ of $${\mathcal{V}(F(p^k))}$$ into $${\mathcal{V}(L_{p,k})}$$ such that ΦΦ( B) = B for every $${B \in \mathcal{V}(L_{p,k})}$$ and ΦΦ( R) = R for every $${R \in \mathcal{V}(F(p^k))}$$. In this paper we show how we can solve an algebraic system of equations over an arbitrary cyclic Post algebra of order p, p prime, using the above interpretation, Gröbner bases and algorithms programmed in Maple. [ABSTRACT FROM AUTHOR]

Boolean and central elements of bounded pseudo-BCK-algebras are studied. Boolean elements form a largest boolean subalgebra and include central elements, which correspond one-one to direct product decompositions. Further, a Cantor-Bernstein type theorem is proved, generalizing similar results for σ-complete MV-algebras and orthogonally σ-complete pseudo-MV-algebras. [ABSTRACT FROM AUTHOR]

Serfati [7] defined a ring structure on every Post algebra. In this paper we determine all the rings that are defined over a Post algebra and share the properties of the Serfati ring. In the case r = 3 one of them is equivalent to the Post algebra. This is a term equivalence and it extends the equivalence between a Boolean algebra and the Boolean ring associated with it. [ABSTRACT FROM AUTHOR]

In a previous paper, we have described all reproductive general solutions of a Post equation, supposing that a general solution is known. In this paper we describe all general solutions of Post equation, supposing that a general solution of this equation is known (Theorem 6). As a special case we get the previous characterization of reproductive solutions and a similar result for Boolean equations (Theorem 9). [ABSTRACT FROM AUTHOR]

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L p,k, and the finite field F( p k) with constants F( p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ1 of the variety V( L p,k) generated by L p,k into the variety V( F( p k)) generated by F( p k) and an interpretation Φ2 of V( F( p k)) into V( L p,k) such that Φ2Φ1( B) = B for every B ε V( L p,k) and Φ1Φ2( R) = R for every R ε V( F( p k)). [ABSTRACT FROM AUTHOR]

It is shown that, if two bounded distributive lattices satisfy the join-infinite distributive law (JID), then their coproduct also satisfies this law. In 1986, Yaqub proved that generalized Post algebras with a finite lattice of constants satisfy JID, and stated that, in general, it is not known whether a generalized Post algebra satisfies JID when its lattice of constants satisfies JID. In this note, the statement is proved. [ABSTRACT FROM AUTHOR]

Abstract: The generalization of the disjunctive and conjunctive Boolean normal forms to fuzzy set theory, obtained by interpreting as a t-norm, as a t-conorm and as a negator, often provides a kind of standard fuzzification procedure. A system of functional equations turns up if some functional independence of the difference between both generalizations is demanded. In this paper, we explore which De Morgan triplets, based on a left-continuous t-norm T, solve this system. Imposing some extra continuity conditions on T, these t-norm solutions can be obtained by the rotation construction of Jenei. The Cauchy equation plays a key role in the reasoning process. [Copyright &y& Elsevier]

The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to σ-complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for these algebras are given. These results are applied to obtain versions of the Cantor-Bernstein-Schröder theorem for σ-complete orthomodular lattices, Stone algebras, BL-algebras, MV-algebras, pseudo MV-algebras, Łukasiewicz and Post algebras of order n. [ABSTRACT FROM AUTHOR]

We define a function d which is a distance in Post algebras. This distance is unique. [Copyright &y& Elsevier]

Deals with the problem of enumeration of Boolean functions in the Post classes F[sup micro][sub 8]. Expression of the problem in terms of the set theory; Notations and proofs of theorems; Definition of a formula for calculating the cardinalities of the Post classes in terms of the graph theory.

The article presents information on MV-algebra and post algebra containing subalgebras of finite orders. Post algebras were introduced by P. C. Rosenbloom and MV algebras by C. C. Chang. Post algebras can be defined as Wajsberg algebras with an additional constant. MV-algebra containing cyclic elements are the elements whose generated subalgebra is a finite MV-chain, i. e. totally ordered with the natural order.

The article discusses the extension method for partial isomorphism and the post-language representation theorem. It attempts to present an analogous result for the first order modal language against a post algebra. It explores the functorial aspect of the first order logic languages and their relevance in the analysis of logical languages.