We consider circuits and expressions whose gates carry out multiplication in a non-associative algebra such as a quasigroup or loop.

In this paper, we characterize exactly the class of languages that are recognizable by finite loops, i.e. by cancellative binary algebras with an identity. This turns out to be the well-studied class of regular open languages. Our proof technique is interesting in itself: we generalize the operation of block product of monoids, which is so useful in the associative case, to the situation where the left factor in the product is non-associative.

Representations of boolean functions as polynomials (over rings) have been used to establish lower bounds in complexity theory. Such representations were used to great effect by Smolensky, who established that MOD q / # AC 0 [MOD p] (for distinct primes p, q) by representing AC 0 [MOD p] functions as low-degree multilinear polynomials over fields of characteristic p. Another tool which has yielded insight into small-depth circuit complexity classes is the program-over-monoids model of computation, which has provided characterizations of circuit complexity classes such as AC 0 and NC 1 .

This paper is organized as follows. Section 2 gives an introduction to the algebraic terms and concepts we will use. In Section 3 we de ne the functional closure of a groupoid, and show that simple non-Abelian loops that are functionally complete. In Section 4 we de ne Boolean-completeness and show that the set of nonBoolean -complete groupoids form a pseudovariety. In Section 5 we de ne the ane quasidirect product and polyabelian loops, and compare polyabelianness to solvability and nilpotence. Sections 6 and 7 are devoted to the proof that polyabelianness correspond precisely to non-Boolean-completeness for loops

The central idea behind the algebraic theory of machines and languages is that recognition by finite state machines and recognition by finite monoids are equivalent. Given a finite state machine M that recognizes a regular language L, each letter of the alphabet is associated with a transformation of the states. Closing these transformations under composition yields a finite monoid called the transformation monoid of M . On an other hand, any finite monoid can be seen as the transformation monoid of some finite state machine. The classification of the regular languages is based on Eilenberg's variety theorem ([6]). A class of languages forms a variety if it is closed under Boolean operations, inverse morphism, and right and left quotients. On an other hand, a class of monoids forms a (pseudo)-variety if it is closed under subgroups, factors, and finite direct products. For example, groups and monoids both for...

An important objective of the algebraic theory of languages is to determine the combinatorial properties of the languages recognized by finite groups and semigroups. In [20], finite nilpotent groups are characterized as those groups that have the ability to count subwords. In this paper, we attempt to generalize this result to finite loops. We introduce the notion of subtree-counting and define subtree-counting loops.