Šare’s algebraic systems

We study algebraic systems MΓ of free semigroup structure, where Γ is a well ordered finite alphabet, discovered in 1970s within the Theory of Electric Circuits by Miro Šare, and and finding recent recent applications in Multivalued Logic, as well as in Computational Linguistics. We provide three simple axioms (reversion axiom (5) and two compression axioms (6) and (7)), which generate the corresponding equivalence relation between words. We also introduce a class of incompressible words, as well as the quotient Šare system MΓ~. The main result is contained in Theorem 16, announced by Šare without proof, which characterizes the equivalence of two words by means of Šare sums. The proof is constructive. We describe an algorithm for compression of words, study homomorphisms between quotient Šare systems for various alphabets Γ (Theorem 38), and introduce two natural Šare categories ŠŠa(M) and ŠŠa(M~). Šare systems are not inverse semigroups.