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...

An effect algebraic partial binary operation $øplus$ defined on the underlying set $E$ uniquely introduces partial order, but not conversely. We show that if on a MacNeille completion $\widehat{E}$ of $E$ there exists an effect algebraic partial binary operation $\widehat{\oplus }$ then $\widehat{\oplus }$ need not be an extension of $\oplus$. Moreover, for an Archimedean atomic lattice effect algebra $E$ we give a necessary and sufficient condition for that $\widehat{\oplus }$ existing on $\widehat{E}$ is an extension of $\oplus$ defined on $E$. Further we show that such $\widehat{\oplus }$ extending $\oplus$ exists at most...

Dually residuated lattice ordered monoids ($DR\ell$-monoids) are common generalizations of, e.g., lattice ordered groups, Brouwerian algebras and algebras of logics behind fuzzy reasonings ($MV$-algebras, $BL$-algebras) and their non-commutative variants ($GMV$-algebras, pseudo $BL$-algebras). In the paper, lex-extensions and lex-ideals of $DR\ell$-monoids (which need not be commutative or bounded) satisfying a certain natural condition are studied.

Bounded commutative residuated lattice ordered monoids ($R\ell$-monoids) are a common generalization of, e.g., $BL$-algebras and Heyting algebras. In the paper, the properties of local and perfect bounded commutative $R\ell$-monoids are investigated.

In analogy with effect algebras, we introduce the test spaces and $MV$-test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between $MV$-algebras and $MV$-test spaces.

If element $z$ of a lattice effect algebra $(E,\oplus ,\mathbf{0},\mathbf{1})$ is central, then the interval $[\mathbf{0},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus$. It is a known fact that if the set $C\left(E\right)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C\left(E\right)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion $\widehat{E}$ of $E$ which is its extension...

Probability on collections of fuzzy sets can be developed as a generalization of the classical probability on $\sigma$-algebras of sets. A Łukasiewicz tribe is a collection of fuzzy sets which is closed under the standard fuzzy complementation and under the pointwise application of the Łukasiewicz t-norm to countably many fuzzy sets. An observable is a fuzzy set-valued mapping defined on a $\sigma$-algebra of sets and satisfying some additional properties; formally, the role of an observable is in a sense analogous...

In the present paper we investigate the relations between maximal completions of lattice ordered groups and maximal completions of pseudo $MV$-algebras.

Bounded residuated lattice ordered monoids ($\mathrm{R}\ell$-monoids) form a class of algebras which contains the class of Heyting algebras, i.e. algebras of the propositional intuitionistic logic, as well as the classes of algebras of important propositional fuzzy logics such as pseudo $\mathrm{MV}$-algebras (or, equivalently, $\mathrm{GMV}$-algebras) and pseudo $\mathrm{BL}$-algebras (and so, particularly, $\mathrm{MV}$-algebras and $\mathrm{BL}$-algebras). Modal operators on Heyting algebras were studied by Macnab (1981), on $\mathrm{MV}$-algebras were studied by Harlenderová and...

Modal operators on Heyting algebras were introduced by Macnab. In this paper we introduce analogously modal operators on MV-algebras and study their properties. Moreover, modal operators on certain derived structures are investigated.

The concept of monadic MV-algebra was recently introduced by A. Di Nola and R. Grigolia as an algebraic formalization of the many-valued predicate calculus described formerly by J. D. Rutledge [9]. This was also genaralized by J. Rachůnek and F. Švrček for commutative residuated $\ell$-monoids since MV-algebras form a particular case of this structure. Basic algebras serve as a tool for the investigations of much more wide class of non-classical logics (including MV-algebras, orthomodular lattices and...

Bounded integral residuated lattices form a large class of algebras containing some classes of commutative and noncommutative algebras behind many-valued and fuzzy logics. In the paper, monotone modal operators (special cases of closure operators) are introduced and studied.

In the paper it is proved that the category of -algebras is equivalent to the category of bounded -semigroups satisfying the identity $1-(1-x)=x$. Consequently, by a result of D. Mundici, both categories are equivalent to the category of bounded commutative -algebras.

The class of commutative dually residuated lattice ordered monoids ($DR\ell$-monoids) contains among others Abelian lattice ordered groups, algebras of Hájek’s Basic fuzzy logic and Brouwerian algebras. In the paper, a unary operation of negation in bounded $DR\ell$-monoids is introduced, its properties are studied and the sets of regular and dense elements of $DR\ell$-monoids are described.