Some Algorithmic Problems for ε-algebras

Abstract

The word problem¹, stated relative to one or another algebraic system, has attracted the attention of many mathematicians. In the works of A.A. Markov [1] and E. Post [3] it was proved for the first time that there exist algebraic systems (semigroups) with undecidable word problem. The most significant achievement in this direction is the result of P.S. Novikov [2] that establishes undecidability of the word problem for groups. In 1950, A.I. Zhukov [5], while studying free nonassociative algebras, established that in the case in which one does not assume that the algebra satisfies any identical relation (for instance, associativity) the word problem (as well as some other algorithmic problems) is decidable. From the results obtained for semigroups, it easily follows that the word problem is undecidable for associative algebras.