# Lattice valued structures, weak congruences, omega-quasigroups and approximate solutions of equations

4th July 2019 the AMU regular meeting took place  at YSU, faculty of Mathematics and Mechanics (room 241).
Prof. Andreja Tepavčević  from the University of Novi Sad, Serbia gave a talk  on the topic “Lattice valued structures, weak congruences, omega-quasigroups and approximate solutions of equations”.
Abstract. The topic of this talk is lattice valued algebras and relational systems, which are classical structures equipped with a generalized (lattice valued) equality replacing the classical “=”. These notions and related techniques originate in universal algebra (with weak congruences), in the general algebra (algebraic structures, quasigroups), in logic (Boolean and Heyting valued models) and in fuzzy mathematics (cut-sets and graded models). Starting with sets where there is a complete lattice, we introduce the notion of an algebra, which is a classical algebra equipped with a valued equality replacing the ordinary one. In these new structures,identities hold as appropriate lattice-theoretic formulas. Identities hold in such an algebra if
and only if they hold on all particular cut-factor algebras, i.e., cut subalgebras over cut-equalities. This approach is directly related with weak congruences of the basic algebra to which a generalized equalityis associated.
That is, every algebra uniquely determines a closure system in the lattice of weak congruences of the basic algebra. By this correspondence we formulate a representation theorem for algebras.Applying general results to omega-quasigroups and related structures, we give answers to existence of approximate solutions of a special type of linear equations with respect to a fuzzy equality, and we describe the solving procedures. Potential applications in coding theory and cryptology are proposed.
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