Modal extensions of Heyting algebras
Modal operators on bounded commutative residuated -monoids
Modal operators on bounded residuated -monoids
Bounded residuated lattice ordered monoids (-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 -algebras (or, equivalently, -algebras) and pseudo -algebras (and so, particularly, -algebras and -algebras). Modal operators on Heyting algebras were studied by Macnab (1981), on -algebras were studied by Harlenderová and...
Modal operators on MV-algebras
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.
Modal operators on symmetrical Heyting algebras
Modal Pseudocomplemented De Morgan Algebras
Modal pseudocomplemented De Morgan algebras (or -algebras for short) are investigated in this paper. This new equational class of algebras was introduced by A. V. Figallo and P. Landini ([Figallo, A. V., Landini, P.: Notes on -valued modal algebras Preprints del Instituto de Ciencias Básicas, Univ. Nac. de San Juan 1 (1990), 28–37.]) and they constitute a proper subvariety of the variety of all pseudocomplemented De Morgan algebras satisfying . Firstly, a topological duality for these algebras...
Modular and Distributive Equalities in Lattices
Modular, distributive and simple intervals of the lattice of topologies
Modulare Polynomverbände über endlichen distributiven Verbänden.
Modularité et distributivité affaibliés dans les lattis
Modularity and distributivity of tolerance lattices of commutative separative semigroups
Molecules and linerly ordered ideals of MV-algebras.
We show that an ideal I of an MV-algebra A is linearly ordered if and only if every non-zero element of I is a molecule. The set of molecules of A is contained in Inf(A) ∪ B2(A) where B2(A) is the set of all elements x ∈ A such that 2x is idempotent. It is shown that I ≠ {0} is weakly essential if and only if B⊥ ⊂ B(A). Connections are shown among the classes of ideals that have various combinations of the properties of being implicative, essential, weakly essential, maximal or prime.
Monadic basic algebras
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 -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...
Monadic involutive pseudo-BCK algebras.
Monadic -valued Łukasiewicz-Moisil algebras
Here we initiate an investigation into the class of monadic -valued Łukasiewicz-Moisil algebras (or -algebras), namely -valued Łukasiewicz-Moisil algebras endowed with a unary operation. These algebras constitute a generalization of monadic -valued Łukasiewicz-Moisil algebras. In this article, the congruences on these algebras are determined and subdirectly irreducible algebras are characterized. From this last result it is proved that is a discriminator variety and as a consequence, the...
Monadic quasi-modal distributive nearlattices
We prove that there is a one to one correspondence between monadic finite quasi-modal operators on a distributive nearlattice and quantifiers on the distributive lattice of its finitely generated filters, extending the results given in ``Calomino I., Celani S., González L. J.: Quasi-modal operators on distributive nearlattices, Rev. Unión Mat. Argent. 61 (2020), 339--352".
Monotone modal operators on bounded integral residuated lattices
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.
Multiplicative lattices and -algebras
MV-algebras are categorically equivalent to a class of -semigroups
In the paper it is proved that the category of -algebras is equivalent to the category of bounded -semigroups satisfying the identity . Consequently, by a result of D. Mundici, both categories are equivalent to the category of bounded commutative -algebras.
MV-algebras with additive closure operators