Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semil... more Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice were first considered by A.V. Figallo, G. Ramón and S. Saad in [11], and independently by S. Celani and D. Montangie in [7]. On the other hand, L. Monteiro introduced the notion of n−valued Hilbert algebras (see [12]). In this work, we investigate the class of n−valued Hilbert algebras with supremum, denoted H ∨ n , i.e., n−valued Hilbert algebras where the associated order is a join-semilattice. The varieties H ∨ n are generated by finite chains. The free H ∨ n −algebra Free n+1 (r) with r generators is studied. In particular, we determine an upper bound to the cardinal of the finitely generated free algebra Free n+1 (r). * This work is dedicated to the memory of professor Guillermina Ramón who was one of the researchers who initiated the study of these algebras.
In this paper we establish a new characterization of 4-valued modal algebras considered by A. Mon... more In this paper we establish a new characterization of 4-valued modal algebras considered by A. Monteiro. In order to obtain this characterization we introduce a new class of algebras named generalized I-algebras. This class contains strictly the class of C-algebras defined by Y. Komori as an algebraic counterpart of the infinite-valued implicative Lukasiewicz propositional calculus. On the other hand, the relationship between I-algebras and conmutative BCK-algebras, defined by S. Tanaka in 1975, allows us to say that in a certain sense G-algebras are also a generalization of these latter algebras.
The first system of many-valued logic was introduced by J. Lukasiewicz, his motivation was of phi... more The first system of many-valued logic was introduced by J. Lukasiewicz, his motivation was of philosophical nature as he was looking for an interpretation of the concepts of possibility and necessity. Since then, plenty of research has been developed in this area. In 1968, when Gr.C. Moisil came across Zadeh's fuzzy set theory he found the motivation he had been looking for in order to legitimate the introduction and study of infinitely-valued Lukasiewicz algebras, so he defined θ−valued Lukasiewicz algebras (or LM θ −algebras, for short) (without negation), where θ is the order type of a chain. In this article, our main interest is to investigate the principal and Boolean congruences and θ−congruences on LM θ −algebras. In order to do this we take into account a topological duality for these algebras obtained in (
Annals of the University of Craiova - Mathematics and Computer Science Series, 2018
In this article we determine a topological duality for M3−lattices, introduced by A. V. Figallo i... more In this article we determine a topological duality for M3−lattices, introduced by A. V. Figallo in the journal Rev. Colombiana de Matem´atica, XXI, 1987 ([3]). By means of this duality we describe the congruences and the subdirectly irreducible M3−lattices and reach some of Figallo’s results in a different way. 2010 Mathematics Subject Classification. Primary 06D75; Secondary 06D30, 03G20. Key words and phrases. M3−lattices, Priestley duality, congruences, subdirectly irreducible algebras.
En este trabajo iniciamos el estudio de una clase de reticulados distributivos, estrechamente rel... more En este trabajo iniciamos el estudio de una clase de reticulados distributivos, estrechamente relacionada con la clase de las algebras de Lukasiewicz trivalentes, cuya consideracion fue sugerida por A. Monteiro [2], [6], [7], [8]. La motivacion para su estudio se encuentra en la teoria de ciertos circuitos electricos. (vease Valentinuzzi [10]).
Communications of the Korean Mathematical Society, 2012
In this article, we continue the study of tense symmetric Heyting algebras (or T SH-algebras). Th... more In this article, we continue the study of tense symmetric Heyting algebras (or T SH-algebras). These algebras constitute a generalization of tense algebras. In particular, we describe a discrete duality for T SH-algebras bearing in mind the results indicated by Or lowska and Rewitzky in [E. Or lowska and I. Rewitzky, Discrete Dualities for Heyting Algebras with Operators, Fund. Inform. 81 (2007), no. 1-3, 275-295] for Heyting algebras. In addition, we introduce a propositional calculus and prove this calculus has T SH-algebras as algebraic counterpart. Finally, the duality mentioned above allowed us to show the completeness theorem for this calculus.
In 2015, A.V. Figallo and G. Pelaitay introduced tense n×m-valued Lukasiewicz-Moisil algebras, as... more In 2015, A.V. Figallo and G. Pelaitay introduced tense n×m-valued Lukasiewicz-Moisil algebras, as a common generalization of tense Boolean algebras and tense n-valued Lukasiewicz-Moisil algebras. Here we initiate an investigation into the class tpLMn×m of tense polyadic n × m-valued Lukasiewicz-Moisil algebras. These algebras constitute a generalization of tense polyadic Boolean algebras introduced by Georgescu in 1979, as well as the tense polyadic n-valued Lukasiewicz-Moisil algebras studied by Chiriţȃ in 2012. Our main result is a representation theorem for tense polyadic n × m-valued Lukasiewicz-Moisil algebras. * The support of CONICET is gratefully acknowledged by Gustavo Pelaitay.
British Journal of Mathematics & Computer Science, 2016
In this note, we introduce the notion of ideal on semi-Heyting algebras which allows us to consid... more In this note, we introduce the notion of ideal on semi-Heyting algebras which allows us to consider a topology on them. Besides, we define the concept of F−multiplier, where F is a topology on a semi-Heyting algebra L, which is used to construct the localization semi-Heyting algebra L F. Furthermore, we prove that the semi-Heyting algebra of fractions L S associated with an ∧−closed system S of L is a semi-Heyting of localization. Finally, in the finite case we prove that L S is isomorphic to a special subalgebra of L. Since Heyting algebras are a particular case of semi-Heyting algebras, all these results generalize those obtained in [11].
J. Vaz De Carvalho and T. Almada in A generalization of the Łukasiewicz algebras, Studia Logica 6... more J. Vaz De Carvalho and T. Almada in A generalization of the Łukasiewicz algebras, Studia Logica 69 (2001), 329-338 introduced the variety L m n , m ≥ 1, n ≥ 2, of m-generalized Łukasiewicz algebras of order n as a generalization of Łukasiewicz algebras of order n and a particular case of Ockham algebras. In this note, we define an implication operation on m-generalized Łukasiewicz algebras of order n, called weak implication, from which we obtain a new description of the congruences on these algebras. This last result allows us to show that L m n is semisimple and locally finite in a different way from that describe in the above mentioned paper.
Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semil... more Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice were first considered by A.V. Figallo, G. Ramón and S. Saad in [11], and independently by S. Celani and D. Montangie in [7]. On the other hand, L. Monteiro introduced the notion of n−valued Hilbert algebras (see [12]). In this work, we investigate the class of n−valued Hilbert algebras with supremum, denoted H ∨ n , i.e., n−valued Hilbert algebras where the associated order is a join-semilattice. The varieties H ∨ n are generated by finite chains. The free H ∨ n −algebra Free n+1 (r) with r generators is studied. In particular, we determine an upper bound to the cardinal of the finitely generated free algebra Free n+1 (r). * This work is dedicated to the memory of professor Guillermina Ramón who was one of the researchers who initiated the study of these algebras.
In this paper we establish a new characterization of 4-valued modal algebras considered by A. Mon... more In this paper we establish a new characterization of 4-valued modal algebras considered by A. Monteiro. In order to obtain this characterization we introduce a new class of algebras named generalized I-algebras. This class contains strictly the class of C-algebras defined by Y. Komori as an algebraic counterpart of the infinite-valued implicative Lukasiewicz propositional calculus. On the other hand, the relationship between I-algebras and conmutative BCK-algebras, defined by S. Tanaka in 1975, allows us to say that in a certain sense G-algebras are also a generalization of these latter algebras.
The first system of many-valued logic was introduced by J. Lukasiewicz, his motivation was of phi... more The first system of many-valued logic was introduced by J. Lukasiewicz, his motivation was of philosophical nature as he was looking for an interpretation of the concepts of possibility and necessity. Since then, plenty of research has been developed in this area. In 1968, when Gr.C. Moisil came across Zadeh's fuzzy set theory he found the motivation he had been looking for in order to legitimate the introduction and study of infinitely-valued Lukasiewicz algebras, so he defined θ−valued Lukasiewicz algebras (or LM θ −algebras, for short) (without negation), where θ is the order type of a chain. In this article, our main interest is to investigate the principal and Boolean congruences and θ−congruences on LM θ −algebras. In order to do this we take into account a topological duality for these algebras obtained in (
Annals of the University of Craiova - Mathematics and Computer Science Series, 2018
In this article we determine a topological duality for M3−lattices, introduced by A. V. Figallo i... more In this article we determine a topological duality for M3−lattices, introduced by A. V. Figallo in the journal Rev. Colombiana de Matem´atica, XXI, 1987 ([3]). By means of this duality we describe the congruences and the subdirectly irreducible M3−lattices and reach some of Figallo’s results in a different way. 2010 Mathematics Subject Classification. Primary 06D75; Secondary 06D30, 03G20. Key words and phrases. M3−lattices, Priestley duality, congruences, subdirectly irreducible algebras.
En este trabajo iniciamos el estudio de una clase de reticulados distributivos, estrechamente rel... more En este trabajo iniciamos el estudio de una clase de reticulados distributivos, estrechamente relacionada con la clase de las algebras de Lukasiewicz trivalentes, cuya consideracion fue sugerida por A. Monteiro [2], [6], [7], [8]. La motivacion para su estudio se encuentra en la teoria de ciertos circuitos electricos. (vease Valentinuzzi [10]).
Communications of the Korean Mathematical Society, 2012
In this article, we continue the study of tense symmetric Heyting algebras (or T SH-algebras). Th... more In this article, we continue the study of tense symmetric Heyting algebras (or T SH-algebras). These algebras constitute a generalization of tense algebras. In particular, we describe a discrete duality for T SH-algebras bearing in mind the results indicated by Or lowska and Rewitzky in [E. Or lowska and I. Rewitzky, Discrete Dualities for Heyting Algebras with Operators, Fund. Inform. 81 (2007), no. 1-3, 275-295] for Heyting algebras. In addition, we introduce a propositional calculus and prove this calculus has T SH-algebras as algebraic counterpart. Finally, the duality mentioned above allowed us to show the completeness theorem for this calculus.
In 2015, A.V. Figallo and G. Pelaitay introduced tense n×m-valued Lukasiewicz-Moisil algebras, as... more In 2015, A.V. Figallo and G. Pelaitay introduced tense n×m-valued Lukasiewicz-Moisil algebras, as a common generalization of tense Boolean algebras and tense n-valued Lukasiewicz-Moisil algebras. Here we initiate an investigation into the class tpLMn×m of tense polyadic n × m-valued Lukasiewicz-Moisil algebras. These algebras constitute a generalization of tense polyadic Boolean algebras introduced by Georgescu in 1979, as well as the tense polyadic n-valued Lukasiewicz-Moisil algebras studied by Chiriţȃ in 2012. Our main result is a representation theorem for tense polyadic n × m-valued Lukasiewicz-Moisil algebras. * The support of CONICET is gratefully acknowledged by Gustavo Pelaitay.
British Journal of Mathematics & Computer Science, 2016
In this note, we introduce the notion of ideal on semi-Heyting algebras which allows us to consid... more In this note, we introduce the notion of ideal on semi-Heyting algebras which allows us to consider a topology on them. Besides, we define the concept of F−multiplier, where F is a topology on a semi-Heyting algebra L, which is used to construct the localization semi-Heyting algebra L F. Furthermore, we prove that the semi-Heyting algebra of fractions L S associated with an ∧−closed system S of L is a semi-Heyting of localization. Finally, in the finite case we prove that L S is isomorphic to a special subalgebra of L. Since Heyting algebras are a particular case of semi-Heyting algebras, all these results generalize those obtained in [11].
J. Vaz De Carvalho and T. Almada in A generalization of the Łukasiewicz algebras, Studia Logica 6... more J. Vaz De Carvalho and T. Almada in A generalization of the Łukasiewicz algebras, Studia Logica 69 (2001), 329-338 introduced the variety L m n , m ≥ 1, n ≥ 2, of m-generalized Łukasiewicz algebras of order n as a generalization of Łukasiewicz algebras of order n and a particular case of Ockham algebras. In this note, we define an implication operation on m-generalized Łukasiewicz algebras of order n, called weak implication, from which we obtain a new description of the congruences on these algebras. This last result allows us to show that L m n is semisimple and locally finite in a different way from that describe in the above mentioned paper.
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Papers by Aldo Figallo