Final Report Summary - PTASNCL (Proof-theoretical and algebraic study of nonclassical logics)
During his Marie Curie IEF project Sándor Jenei has considered the study of different substructural logics based on his geometric approach. His main results are listed below.
Uninorms are becoming important algebraic structures in non-classical logics. By analogy with the usual extension of the group operation from the positive cone of an ordered Abelian group into the whole group, a construction - called symmetrization - is defined and it is related the well-known rotation construction of Jenei in the paper entitled "On the relationship between the rotation construction and ordered Abelian groups". Symmetrization is shown to be a kind of dualized rotation. A characterization is given for the left-continuous t-conorms for which their symmetrization is a uninorm. As a by-product a new family of involutive uninorms is introduced. The paper has appeared in Fuzzy Sets and Systems.
In the paper entitled "Structural description of involutive uninorms and finite uninorm chains via skew symmetrization" S. Jenei presented a complete structural description for the class of e-involutive uninorms, which is a particular class of residuated lattices on [0,1], and is closely related to left-continuous t-norms and t-conorms. In addition, he introduced a new construction, called skew symmetrization, which results in e-involutive uninorms. The main theorem of the paper states that every e-involutive uninorm on [0,1] can be described as the skew-symmetrization of its underlying t-norm or its underlying conorm. Moreover, skew symmetrization has been shown to be an unorthodox and unusual generalization of the well-known cone-representation of ordered groups for the whole class of residuated lattices. The meta-mathematical result of the paper is to point out that for the structural description of residuated structures one has to employ co-residuation too. This is a surprising observation in the theory of residuated lattices, a theory that goes back to 70 years. The paper has appeared in the prestigious Journal of Logic and Computation.
In the paper entitled "On reflection invariance of residuated chains" it is shown that under certain conditions, a subset of the graph of a commutative residuated chain is invariant under a geometric reflection. This result implies that a certain part of the graph of the monoidal operation of a commutative residuated chain determines another part of the graph via the reflection on one hand, and tells us about the structure of continuity points of the monoidal operation on the other. The paper has appeared in the prestigious Annals of Pure and Applied Logic.
In the paper entitled "On Involutive FLe-algebras" (joint paper with Hiroakira Ono) S. Jenei investigates involutive uninorms. These operations are important in certain substructural logics. Beyond a general structural description for the conic case, the finite chain case has been investigated in details.
In the paper entitled "Equality Algebras" S. Jenei introduces a new algebraic structure, called equality algebras, and relates this structure to BCK-algebras with meet. As a side-result, a result of Kabzin´ski, Wron´ski about Heyting algebras is generalized: An equational characterization is presented for the equivalential fragment of BCK-algebras with meet.
Uninorms are becoming important algebraic structures in non-classical logics. By analogy with the usual extension of the group operation from the positive cone of an ordered Abelian group into the whole group, a construction - called symmetrization - is defined and it is related the well-known rotation construction of Jenei in the paper entitled "On the relationship between the rotation construction and ordered Abelian groups". Symmetrization is shown to be a kind of dualized rotation. A characterization is given for the left-continuous t-conorms for which their symmetrization is a uninorm. As a by-product a new family of involutive uninorms is introduced. The paper has appeared in Fuzzy Sets and Systems.
In the paper entitled "Structural description of involutive uninorms and finite uninorm chains via skew symmetrization" S. Jenei presented a complete structural description for the class of e-involutive uninorms, which is a particular class of residuated lattices on [0,1], and is closely related to left-continuous t-norms and t-conorms. In addition, he introduced a new construction, called skew symmetrization, which results in e-involutive uninorms. The main theorem of the paper states that every e-involutive uninorm on [0,1] can be described as the skew-symmetrization of its underlying t-norm or its underlying conorm. Moreover, skew symmetrization has been shown to be an unorthodox and unusual generalization of the well-known cone-representation of ordered groups for the whole class of residuated lattices. The meta-mathematical result of the paper is to point out that for the structural description of residuated structures one has to employ co-residuation too. This is a surprising observation in the theory of residuated lattices, a theory that goes back to 70 years. The paper has appeared in the prestigious Journal of Logic and Computation.
In the paper entitled "On reflection invariance of residuated chains" it is shown that under certain conditions, a subset of the graph of a commutative residuated chain is invariant under a geometric reflection. This result implies that a certain part of the graph of the monoidal operation of a commutative residuated chain determines another part of the graph via the reflection on one hand, and tells us about the structure of continuity points of the monoidal operation on the other. The paper has appeared in the prestigious Annals of Pure and Applied Logic.
In the paper entitled "On Involutive FLe-algebras" (joint paper with Hiroakira Ono) S. Jenei investigates involutive uninorms. These operations are important in certain substructural logics. Beyond a general structural description for the conic case, the finite chain case has been investigated in details.
In the paper entitled "Equality Algebras" S. Jenei introduces a new algebraic structure, called equality algebras, and relates this structure to BCK-algebras with meet. As a side-result, a result of Kabzin´ski, Wron´ski about Heyting algebras is generalized: An equational characterization is presented for the equivalential fragment of BCK-algebras with meet.