Bounded commutative residuated lattice ordered monoids ($R\ell$-monoids) are a common generalization of $\mathrm{𝐵𝐿}$-algebras and Heyting algebras, i.e. algebras of basic fuzzy logic and intuitionistic logic, respectively. In the paper we develop the theory of filters of bounded commutative $R\ell$-monoids.

The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (monoids) are common generalizations of $\text{BL}$-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding...

The concept of a commutative directoid was introduced by J. Ježek and R. Quackenbush in 1990. We complete this algebra with involutions in its sections and show that it can be converted into a certain implication algebra. Asking several additional conditions, we show whether this directoid is sectionally complemented or whether the section is an NMV-algebra.

A subobjects structure of the category $\Omega$- of $\Omega$-fuzzy sets over a complete $MV$-algebra $\Omega =(L,\wedge ,\vee ,\otimes ,\to )$ is investigated, where an $\Omega$-fuzzy set is a pair $𝐀=(A,\delta )$ such that $A$ is a set and $\delta A\times A\to \Omega$ is a special map. Special subobjects (called complete) of an $\Omega$-fuzzy set $𝐀$ which can be identified with some characteristic morphisms $𝐀\to {\Omega }^{*}=(L\times L,\mu )$ are then investigated. It is proved that some truth-valued morphisms ${\neg }_{\Omega }{\Omega }^{*}\to {\Omega }^{*},{\cap }_{\Omega }$, ${\cup }_{\Omega }{\Omega }^{*}\times {\Omega }^{*}\to {\Omega }^{*}$ are characteristic morphisms of complete subobjects.

In this paper we construct conditional states on semi-simple MV-algebras. We show that these conditional states are not given uniquely. By using them we construct the joint probability distributions and discuss the properties of these distributions. We show that the independence is not symmetric.

We deal with congruences on semilattices with section antitone involution which rise e.g., as implication reducts of Boolean algebras, MV-algebras or basic algebras and which are included among implication algebras, orthoimplication algebras etc. We characterize congruences by their kernels which coincide with semilattice filters satisfying certain natural conditions. We prove that these algebras are congruence distributive and 3-permutable.

We generalize the correspondence between basic algebras and lattices with section antitone involutions to a more general case where no lattice properties are assumed. These algebras are called conjugated if this correspondence is one-to-one. We get conditions for the conjugary of such algebras and introduce the induced relation. Necessary and sufficient conditions are given to indicated when the induced relation is a quasiorder which has “nice properties", e.g. the unary operations are antitone...

MV-algebras were introduced in 1958 by Chang [4] and they are models of Lukasiewicz infinite-valued logic. Chang gives a correspondence between the category of linearly ordered MV-algebras and the category of linearly ordered abelian l-groups.Mundici [10] extended this result showing a categorical equivalence between the category of the MV-algebras and the category of the abelian l-groups with strong unit.In this paper, starting from some definitions and results in abelian l-groups, we shall study...

We obtain for measures on MV-algebras the classical theorem of Dieudonné related to convergent sequences of regular maps.

For a pseudo $MV$-algebra $𝒜$ we denote by $\ell \left(𝒜\right)$ the underlying lattice of $𝒜$. In the present paper we investigate the algebraic properties of maximal convex chains in $\ell \left(𝒜\right)$ containing the element 0. We generalize a result of Dvurečenskij and Pulmannová.