Functional equations with division and regular operations

Functional equations are equations in which the unknown (or unknowns) are functions. We consider equations of generalized associativity, mediality (bisymmetry, entropy), paramediality, transitivity as well as the generalized Kolmogoroff equation. Their usefulness was proved in applications both in mathematics and in other disciplines, particularly in economics and social sciences (see J. Aczél, On mean values, Bull. Amer. Math. Soc. 54 (1948) 392–400; J. Aczél, Remarques algebriques sur la solution donner par M. Frechet a l’equation de Kolmogoroff, Pupl. Math. Debrecen 4 (1955) 33–42; J. Aczél, A Short Course on Functional Equations Based Upon Recent Applications to the Social and Behavioral Sciences, Theory and decision library, Series B: Mathematical and statistical methods (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987); J. Aczél, Lectures on Functional Equations and Their Applications (Supplemented by the author, ed. H. Oser) (Dover Publications, Mineola, New York, 2006); J. Aczél, V. D. Belousov and M. Hosszu, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar. 11 (1960) 127–136; J. Aczél and J. Dhombres, Functional Equations in Several Variables (Cambridge University Press, New York, 1991); J. Aczél and T. L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psych. 27(1) (1983) 93–102). We use unifying approach to solve these equations for division and regular operations generalizing the classical quasigroup case.

Aggregating Fuzzy QL- and (S,N)-Subimplications: Conjugate and Dual Constructions

<p>Fuzzy (S,N)- and <span style="text-decoration: underline;">QL</span>-<span style="text-decoration: underline;">subimplication</span> classes can be obtained by a distributive n-<span style="text-decoration: underline;">ry</span> aggregation performed over the families of t-<span style="text-decoration: underline;">subnorms</span> and t-<span style="text-decoration: underline;">subconorms</span> along with a fuzzy negation. Since these classes of <span style="text-decoration: underline;">subimplications</span> are explicitly represented by t-<span style="text-decoration: underline;">subconorms</span> and t-<span style="text-decoration: underline;">subnorms</span> verifying the generalized associativity, the corresponding (S,N)- and <span style="text-decoration: underline;">QL</span>-<span style="text-decoration: underline;">subimplications</span>, referred as I(S,N) and I_(S,T,N), are characterized as distributive n-<span style="text-decoration: underline;">ary</span> aggregation together with related generalizations as the exchange and neutrality principles. Based on these results, the both subclasses I_(S,n) and I_QL of (S,N)- and <span style="text-decoration: underline;">QL</span>-<span style="text-decoration: underline;">subimplications</span> which are obtained by the median aggregation operation performed over the standard negation N_S together with the families of t-<span style="text-decoration: underline;">subnorms</span> and t-<span style="text-decoration: underline;">subconorms</span> S_P and T_P, respectively. In particular, the subclass T_P extends the product t-norm T_P as well as S_P extends the algebraic sum S_P. As the main results, the family of <span style="text-decoration: underline;">subimplications</span> I_(S_P,N) and I_(S_P,T_P,N) extends the implication class by preserving the corresponding properties. We also present an extension from (S,N)- and <span style="text-decoration: underline;">QL</span>-<span style="text-decoration: underline;">subimplications</span> to (S,N)- and <span style="text-decoration: underline;">QL</span>-implications and discuss dual and conjugate constructions.</p><pre><!--EndFragment--></pre>