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It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relatively residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relatively residuated lattices which are similar to those known for residuated ones and extend our results to posets.

The so-called basic algebras correspond in a natural way to lattices with antitone involutions and hence generalize both MV-algebras and orthomodular lattices. The paper deals with several types of special elements of basic algebras and with pseudocomplemented basic algebras.

The aim of the paper is to describe the connections between lattice-ordered commutative loops and certain "basic algebras" which are a non-associative generalization of MV-algebras and are related to commutative semicopulas. This extends the well-known equivalence between lattice-ordered Abelian groups with strong order-unit and MV-algebras.

A number of research articles have established the significant role of lattice-ordered groups (l-groups) in logic. The fact that underpins these studies is the realization that important algebras of logic may be viewed as l-groups with a modal operator. These connections are just the tip of the iceberg. The purpose of the present article is to lay the groundwork for, and provide significant initial contributions to, the development of a Conrad type approach to the study of algebras of logic. The term Conrad Program refers to Paul Conrad's approach to the study of l-groups, which analyzes the structure of individual or classes of l-groups by primarily using strictly lattice theoretic properties of their lattices of convex l-subgroups. The present article demonstrates that large parts of the Conrad Program can be profitably extended in the setting of e-cyclic residuated lattices – that is residuated lattices that satisfy the identity x \ e ≈ e / x . An indirect benefit of this work is the introduction of new tools and techniques in the study of algebras of logic, and the enhanced role of the lattice of convex subalgebras of a residuated lattice.

Pseudo-BCI-algebras generalize both BCI-algebras and pseudo-BCK-algebras, which are a non-commutative generalization of BCK-algebras. In this paper, following [J.G. Raftery and C.J. van Alten, Residuation in commutative ordered monoids with minimal zero, Rep. Math. Log. 34 (2000) 23-57], we show that pseudo-BCI-algebras are the residuation subreducts of semi-integral residuated po-monoids and characterize those pseudo-BCI-algebras which are direct products of pseudo-BCK-algebras and groups (regarded as pseudo-BCI-algebras). We also show that the quasivariety of pseudo-BCI-algebras is relatively congruence modular; in fact, we prove that this holds true for all relatively point regular quasivarieties which are relatively ideal determined, in the sense that the kernels of relative congruences can be described by means of ideal terms.

We axiomatize the smallest variety that contains both the variety of MV-algebras and the variety (term equivalent to the variety) of orthomodular lattices.

Distributive lattices with antitone involutions (or equivalently, distributive basic algebras) are studied. It is proved that in the finite case their underlying lattices are isomorphic to direct products of finite chains, and hence finite distributive basic algebras can be constructed by "perturbing" finite MV-algebras, and moreover, under certain natural conditions, they even coincide with finite MV-algebras. Sharp elements in basic algebras satisfying these natural conditions are studied, too.

The paper is based on the authors’ talks given at the 11th IQSA Meeting in Cagliari; it deals with states and state operators (internal states) on basic algebras which are a generalization of MV-algebras and orthomodular lattices, also including lattice effect algebras. On the one hand, the paper is a survey of some previous results on states and state operators on commutative basic algebras, and on the other one, an extension of these results to the much larger class of basic algebras where the addition ⊕ distributes over the lattice meet ∧.

Lattice-ordered effect algebras generalize both MV-algebras and orthomodular lattices. In this paper, finitely generated varieties of distributive lattice effect algebras are axiomatized, and for any positive integer n, the free n-generator algebras in these varieties are described.

A common generalization of effect algebras and ortholattices that allows to represent ortholattices in a similar way in which orthomodular lattices are represented in the setting of effect algebras is introduced.

MV-algebras as well as orthomodular lattices can be seen as a particular case of so-called "basic algebras" which are an alter ego of bounded lattices whose sections are equipped with fixed antitone involutions. The class of basic algebras is an ideal variety. In the paper, we give an internal characterization of congruence kernels (ideals) and find a finite basis of ideal terms, with focus on monotone and effect basic algebras. We also axiomatize basic algebras that are subdirect products of linearly ordered ones.

The paper deals with states on commutative basic algebras that are a non-associative generalization of MV-algebras or, in other words, the algebraic semantics for a fuzzy logic which generalizes the Lukasiewicz logic in that the conjunction is not associative. States are defined in the same way as Mundici's states on MV-algebras as normalized finitely additive [0,1]-valued functions, and some results analogous to the results that are known from MV-algebras are proved.

Basic algebras are a generalization of MV-algebras, also including orthomodular lattices and lattice effect algebras. A pre-ideal of a basic algebra is a non-empty subset that is closed under the addition ⊕ and downwards closed with respect to the underlying order. In this paper, we study the pre-ideal lattices of algebras in a particular subclass of basic algebras which are closer to MV-algebras than basic algebras in general. We also prove that finite members of this subclass are exactly finite MV-algebras.

For each n, the free abelian group F on countably many free generators will be equipped with a monoid P n making (F,P n) into the free n-generator abelian ℓ-group. This is a generalization of the main result of the second author, published in the J. Pure Appl. Algebra 208 (2007), 549–554, only dealing with the case n = 2.

Double basic algebras are a counterpart of bounded lattices with order-antiautomorphisms on principal filters. In the paper, an independent axiomatization of double basic algebras is given and lattice pseudo-effect algebras are characterized in the setting of double basic algebras.

It is shown that an arbitrary interval of a pseudo-effect algebra is a pseudo-effect algebra and some results concerning Riesz decomposition properties, compatibilities and states are proved.

Pseudo-BCK-algebras are a non-commutative generalization of well-known BCK-algebras. The paper describes a situation when a linearly ordered pseudo-BCK-algebra is an ordinal sum of linearly ordered cone algebras. In addition, we present two identities giving such a possibility of the decomposition and axiomatize the residuation subreducts of representable pseudo-hoops and pseudo-BL-algebras.

We prove that every effect algebra (E,+, 0, 1) can easily be made into a total algebra (E,⊕, ¬, 0) of type (2, 1, 0) in such a way that two elements are compatible in (E,+, 0, 1) if and only if they commute in (E,⊕, ¬, 0).

It is shown that every pseudo-effect algebra can be organized as a total algebra. In general, this total algebra is not unique, but it always determines the original partial addition. Special attention is paid to lattice-ordered pseudo-effect algebras for which the total operations can be defined in a unique natural way and this allows one to characterize pairs of compatible elements.

We generalize the concept of an integral residuated lattice to join-semilattices with an upper bound where every principal order-filter (section) is a residuated semilattice; such a structure is called a sectionally residuated semilattice. Natural examples come from propositional logic. For instance, implication algebras (also known as Tarski algebras), which are the algebraic models of the implication fragment of the classical logic, are sectionally residuated semilattices such that every section is even a Boolean algebra. A similar situation rises in case of the Lukasiewicz multiple-valued logic where sections are bounded commutative BCK-algebras, hence MV-algebras. Likewise, every integral residuated (semi)lattice is sectionally residuated in a natural way. We show that sectionally residuated semilattices can be axiomatized as algebras (A, r, →, ⇝, 1) of type 〈3, 2, 2, 0〉 where (A, →, ⇝, 1) is a {→, ⇝, 1}-subreduct of an integral residuated lattice. We prove that every sectionally residuated lattice can be isomorphically embedded into a residuated lattice in which the ternary operation r is given by r(x, y, z) = (x · y) ∨ z. Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras.

We deal with algebras \({\bf A} = (A, \oplus, \neg, 0)\) of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) A bears a natural lattice structure, (ii) the elements a for which \(\neg a\) is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety.

We deal with unbounded dually residuated lattices that generalize pseudo $MV$-algebras in such a way that every principal order-ideal is a pseudo $MV$-algebra. We describe the connections of these generalized pseudo $MV$-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo $MV$-algebra $A$ by means of the positive cone of a suitable $\ell $-group $G_A$. We prove that the lattice of all (normal) ideals of $A$ and the lattice of all (normal) convex $\ell $-subgroups of $G_A$ are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo $MV$-algebra is commutative.

In this paper de Finetti's (no-Dutch-Book) criterion for coherent probability assignments is extended to large classes of logics and their algebras. Given a set A of ''events'' and a closed set W@?[0,1]^A of ''possible worlds'' we show that a map s:A->[0,1] satisfies de Finetti's criterion if, and only if, it has the form s(a)=@!"WV(a)d@m(V) for some probability measure @m on W. Our results are applicable to all logics whose connectives are continuous operations on [0,1], notably (i) every [0,1]-valued logic with finitely many truth-values, (ii) every logic whose conjunction is a continuous t-norm, and whose negation is @?x=1-x, possibly also equipped with its t-conorm and with some continuous implication, (iii) any extension of Lukasiewicz logic with constants or with a product-like connective. We also extend de Finetti's criterion to the noncommutative underlying logic of GMV-algebras.

Pseudo-BCK-algebras arise as the { \ , / , 1 } -subreducts of integral residuated lattices. In this note we characterize pseudo-BCK-algebras that are subdirect products of linearly ordered pseudo-BCK-algebras.

We consider a non-associative generalization of MV-algebras. The underlying posets of our non-associative MV-algebras are not lattices, but they are related to so-called λ-lattices.

We prove that if A and B are orthogonally σ-complete commutative pseudo-BCK-algebras such that A is isomorphic to a direct factor in B, and also B is isomorphic to a direct factor in A, then A and B are isomorphic. As a consequence we obtain previously known results for MV-algebras (by De Simone, Mundici and Navara), pseudo-MV-algebras (by Jakubik) and lattice-ordered groups (again by Jakubik).

Sectionally pseudocomplemented semilattices are an extension of relatively pseudocomplemented semilattices—they are meet-semilattices with a greatest element such that every section, i.e., every principal filter, is a pseudocomplemented semilattice. In the paper, we give a simple equational characterization of sectionally pseudocomplemented semilattices and then investigate mainly their congruence kernels which leads to a characterization of subdirectly irreducible sectionally pseudocomplemented semilattices.

In this paper, we introduce the concept of an ideal of a noncommutative dually residuated lattice ordered monoid and we show that congruence relations and certain ideals are in a one-to-one correspondence.

We set up an axiomatic system for the logical connective implication within the framework of MV-algebras which generalizes implication in classical logic described similarly by J. C. Abbott. The induced structure (weak implication algebra) turns to be a join-semilattice whose principal filters are MV-algebras.

Dually residuated lattice-ordered monoids ($DR\ell $-monoids for short) generalize lattice-ordered groups and include for instance also $GMV$-algebras (pseudo $MV$-algebras), a non-commutative extension of $MV$-algebras. In the present paper, the spectral topology of proper prime ideals is introduced and studied.

It is shown that pseudo $BL$-algebras are categorically equivalent to certain bounded $DR\ell $-monoids. Using this result, we obtain some properties of pseudo $BL$-algebras, in particular, we can characterize congruence kernels by means of normal filters. Further, we deal with representable pseudo $BL$-algebras and, in conclusion, we prove that they form a variety.