By Masami Ito, Teruo Imaoka

The study effects released during this booklet diversity from natural mathematical conception (semigroup idea, discrete arithmetic, etc.) to theoretical computing device technological know-how, specifically formal languages and automata. The papers handle matters within the algebraic and combinatorial theories of semigroups, phrases and languages, the constitution concept of automata, the category idea of formal languages and codes, and functions of those theories to numerous components, like quantum and molecular computing, coding concept, and cryptography.

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Which characterize the synonymy classes. Since the above definitions were given in a very general setting, we should restrict our investigation to particular valuations. To this end, in this paper we shall only consider the valuations in the ring of integers Z with the usual addition and multiplication. Of course, other semirings may be considered as well, 27 but we have chosen this semiring because it is the most natural and simple one. Even so, the problems we considered appeared t o be very difficult.

XWy,xW) (4) with i E I , x a variable, and the y j (w - 1)-words of height at most 1 which are 1 in groups. Proof. Since V 2 S1, the local semillatice B2 belongs to C V . Note that C V is defined by the pseudoidentities of the form (4)where x , y1 , , .. ,Y n are distinct variables and u i , q depend on the same n variables. 24, g C V is defined by the pseudoidentities of the form (4) viewed over the corresponding most general graph supporting both sides. But, for the variables x,yk the equivalence relation defining r identifies x with 2-l and y;', and also identifies Y k with x-'.

Ash, Finite semigroups with commuting idempotents, J . Austral. Math. ,Ser. A 43 (1987) 81-90. ~ 3 1 , Inevitable graphs: a proof of the type 11 conjecture and some related decision procedures, Int. J. Algebra and Computation 1 (1991) 127-146. [14] K. Auinger e B. Steinberg, T h e geometry of profinite graphs with applications to free groups and finite monoids, Tech. Rep. CMUP 2001-06, 2001. [15] K. Auinger and B. Steinberg, O n the extension problem f o r partial permutations, Tech. Rep. CMUP 2001-08, 2001.

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