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Given an n-ary k-valued function f, gap(f) denotes the essential arity gap of f which is the minimal number of essential variables in f
which become fictive when identifying any two distinct essential variables in f.
In the present paper we study the properties of the symmetric function
with non-trivial arity gap (2 ≤ gap(f)). We prove several results concerning
decomposition of the symmetric functions with non-trivial arity gap with
its minors or subfunctions. We show that all non-empty sets of...
Four-part semigroups form a new class of semigroups which became important when sets of Boolean operations which are closed under the binary superposition operation f + g := f(g,...,g), were studied. In this paper we describe the lattice of all subsemigroups of an arbitrary four-part semigroup, determine regular and idempotent elements, regular and idempotent subsemigroups, homomorphic images, Green's relations, and prove a representation theorem for four-part semigroups.
The general Galois theory for functions and relational constraints over arbitrary sets described in the authors' previous paper is refined by imposing algebraic conditions on relations.
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