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Implicative hyper K -algebras

Mohammad Mehdi Zahedi, A. Borumand Saeid, R. A. Borzooei (2005)

Czechoslovak Mathematical Journal

In this note we first define the notions of (weak, strong) implicative hyper K -algebras. Then we show by examples that these notions are different. After that we state and prove some theorems which determine the relationship between these notions and (weak) hyper K -ideals. Also we obtain some relations between these notions and (weak) implicative hyper K -ideals. Finally, we study the implicative hyper K -algebras of order 3, in particular we obtain a relationship between the positive implicative...

Interior and closure operators on bounded commutative residuated l-monoids

Jiří Rachůnek, Filip Švrček (2008)

Discussiones Mathematicae - General Algebra and Applications

Topological Boolean algebras are generalizations of topological spaces defined by means of topological closure and interior operators, respectively. The authors in [14] generalized topological Boolean algebras to closure and interior operators of MV-algebras which are an algebraic counterpart of the Łukasiewicz infinite valued logic. In the paper, these kinds of operators are extended (and investigated) to the wide class of bounded commutative Rl-monoids that contains e.g. the classes of BL-algebras...

Interior and closure operators on bounded residuated lattice ordered monoids

Filip Švrček (2008)

Czechoslovak Mathematical Journal

G M V -algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior G M V -algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on D R l -monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on G M V -algebras.

Interior and Closure Operators on Commutative Bounded Residuated Lattices

Jiří Rachůnek, Zdeněk Svoboda (2013)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Commutative bounded integral residuated lattices form a large class of algebras containing some classes of algebras behind many valued and fuzzy logics. In the paper we introduce and investigate additive closure and multiplicative interior operators on this class of algebras.

Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics

Larisa Maksimova (1999)

Banach Center Publications

We consider the families 𝓛 of propositional superintuitionistic logics (s.i.l.) and NE(K) of normal modal logics (n.m.l.). It is well known that there is a duality between 𝓛 and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also NE(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig's interpolation property (CIP), the disjunction property (DP), the Beth property (BP), Hallden-completeness...

Isometries of generalized M V -algebras

Ján Jakubík (2007)

Czechoslovak Mathematical Journal

In this paper we investigate the relations between isometries and direct product decompositions of generalized M V -algebras.

Join-semilattices whose sections are residuated po-monoids

Ivan Chajda, Jan Kühr (2008)

Czechoslovak Mathematical Journal

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...

Letter to the editor: Consistency of LPC+Ch

Jorma K. Mattila (1998)

Kybernetika

In his paper [Kybernetika 31, No. 1, 99–106 (1995; Zbl 0857.03042)], E. Turunen says in the corollary on p. 106: “Notice that the third last line on page 195 in [J. K. Mattila, “Modifier logic”, in: J. Kacprzyk (ed.) et al., Fuzzy logic for the management of uncertainty. New York: Wiley. 191–209 (1992)] stating that LPC+Ch calculus is consistent is not correct.” The system LPC+Ch is consistent, which can be seen quite trivially.

Many-sorted coalgebraic modal logic : a model-theoretic study

Bart Jacobs (2001)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many-sorted Boolean Algebras with...

Many-Sorted Coalgebraic Modal Logic: a Model-theoretic Study

Bart Jacobs (2010)

RAIRO - Theoretical Informatics and Applications

This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many-sorted Boolean Algebras with...

Molecules and linerly ordered ideals of MV-algebras.

C. S. Hoo (1997)

Publicacions Matemàtiques

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

Ivan Chajda, Miroslav Kolařík (2008)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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 ℓ -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 quasi-modal distributive nearlattices

Ismael Calomino (2023)

Commentationes Mathematicae Universitatis Carolinae

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

Jiří Rachůnek, Zdeněk Svoboda (2012)

Mathematica Bohemica

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.

New kinds of hybrid filters of EQ-algebras

Akbar Paad (2024)

Mathematica Bohemica

The main goal of this paper is to introduce hybrid positive implicative and hybrid implicative (pre)filters of EQ-algebras. In the following, some characterizations of this hybrid (pre)filters are investigated and it is proved that the quotient algebras induced by hybrid positive implicative filters in residuated EQ-algebras are idempotent and residuated EQ-algebra. Moreover, the relationship between hybrid implicative prefilters and hybrid positive implicative prefilters are discussed and it is...

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