Esko Turunen

States on semi-divisible residuated lattices

Given a residuated lattice L, we prove that the subset MV(L) of complement elements x* of L generates an MV-algebra if, and only if L is semi-divisible. Riečan states on a semi-divisible residuated lattice L, and Riečan states on MV(L) are essentially the very same thing. The same holds for Bosbach states as far as L is divisible. There are semi-divisible residuated lattices that do not have Bosbach states.

States on semi-divisible generalized residuated lattices reduce to states on MV-algebras

A semi-divisible residuated lattice is a residuated lattice L satisfying an additional condition weaker than that of divisibility. Such structures are related to mathematical fuzzy logic as well as to extended probability theory by the fact that the subset of complemented elements induces an MV-algebra. We define generalized residuated lattices by omitting commutativity of the corresponding monoidal operation and study semi-divisibility in such structures. We show that, given a good generalized residuated lattice L, the set of complemented elements of L, denoted by MV(L), forms a pseudo-MV-algebra if and only if L is semi-divisible. Maximal filters on a semi-divisible generalized residuated lattice L are in one-to-one correspondence with maximal filters on MV(L). We study states on semi-divisible generalized residuated lattices. Riecan states on a semi-divisible generalized residuated lattice L are determined by Riecan states on MV(L). The same holds true for Bosbach states whenever L is a good divisible generalized residuated lattice. Extremal Riecan states on a semi-divisible generalized residuated lattice L are in one-to-one correspondence with maximal and semi-normal filters on L.

A characterization of fuzzy implications generated by generalized quantifiers

We characterize particular fuzzy implications, called special implication connectives, which are generated by certain general unary hypothesis automaton (GUHA for short) data mining quantifiers, called special implicational quantifiers.

n -Fold implicative basic logic is Gödel logic

We prove that Haveshki’s and Eslami’s n-fold implicative basic logic is Gödel logic and n-fold positive implicative basic logic is a fragment of ukasiewicz logic.

Another paraconsistent algebraic semantics for Lukasiewicz–Pavelka logic

Abstract As recently proved in a previous work of Turunen, Tsoukias and Ozturk, starting from an evidence pair ( a , b ) on the real unit square and associated with a propositional statement α, we can construct evidence matrices expressed in terms of four values t, f, k, u that respectively represent the logical valuations true , false , contradiction (both true and false) and unknown (neither true nor false) regarding the statement α. The components of the evidence pair ( a , b ) are to be understood as evidence for and against α, respectively. Moreover, the set of all evidence matrices can be equipped with an injective MV-algebra structure. Thus, the set of evidence matrices can play the role of truth-values of a Lukasiewicz–Pavelka fuzzy logic, a rich and applicable mathematical foundation for fuzzy reasoning, and in such a way that the obtained new logic is paraconsistent. In this paper we show that a similar result can be also obtained when the evidence pair ( a , b ) is given on the real unit triangle. Since the real unit triangle does not admit a natural MV-structure, we introduce some mathematical results to show how this shortcoming can be overcome, and another injective MV-algebra structure in the corresponding set of evidence matrices is obtained. Also, we derive several formulas to explicitly calculate the evidence matrices for the operations associated to the usual connectives.

Minimal solutions of general fuzzy relation equations on linear carriers. An algebraic characterization

Abstract This paper considers a general fuzzy relation equation, which has minimal solutions, if it is solvable. In this case, an algebraic characterization is introduced which provides an interesting method to compute minimal solutions in this general setting. Moreover, a comparison with other frameworks is also given.

Hyper-Archimedean BL-algebras are MV-algebras

Generalizations of Boolean elements of a BL-algebra L are studied. By utilizing the MV-center MV(L) of L, it is reproved that an element x L is Boolean iff x x * = 1. L is called semi-Boolean if for all x L, x * is Boolean. An MV-algebra L is semi-Boolean iff L is a Boolean algebra. A BL-algebra L is semi-Boolean iff L is an SBL-algebra. A BL-algebra L is called hyper-Archimedean if for all x L, xn is Boolean for some finite n 1. It is proved that hyper-Archimedean BL-algebras are MV-algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV-algebras or BL-algebras. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Survey of Theory and Applications of Łukasiewicz-Pavelka Fuzzy Logic

We demonstrate how approximate reasoning, many classification tasks, case-based reasoning, etc. can be viewed as applications of many valued similarity and, thus Lukasiewicz-Pavelka logic.

Local MV-Algebras

Local MV-algebras are MV-algebras with only one maximal ideal that, hence, contains all infinitesimal elements.

Soft Computing Methods

Soft computing methods of modelling usually include fuzzy logics , neural computation , genetical algorithms and probabilistic deduction , with the addition of data mining and chaos theory in some cases. Unlike the traditional “hardcore methods” of modelling, these new methods allow for the gained results to be incomplete or inexact. Methodologically, the different approaches of these soft methods are quite heterogeneous. Still, all of them have a few things in common, namely that they have all been developed during the last 30–50 years (Bayes formula in 1763 and Lukasiewicz logic in 1920 being the exceptions), and that they would probably have not achieved their current standards without the exceptional growth in computational capacities of computers.