No abstract

No abstract

Abstract. There are several known ways to define a product automaton on the cartesian product of the state sets of two given automata. This paper introduces a new product called the cartesian composition and discusses how various properties of the product automaton depend on the corresponding properties of the factors. A main result is that any finite connected automaton has a unique representation as a cartesian composition of prime automata.In the Algebraic Theory of Automata several definitions of the product of automata are known. One type of products defines the product-automaton on the cartesian product of the state-sets of the given automata. In the direct product, one can fix the input-set or take the cartesian product of the input sets as the new input set. In this context the reader is referred to [1] where these two kinds of products are distinguished as homogeneous and heterogeneous product. The first possibility has been particularly studied in several papers [5,6,8,9,11]. The reader is referred to [7] where an almost complete bibliography can be found, for almost all definitions and notations used in the following and for related results. We list [4] as a reference on the general theory of automata. Our task will be to introduce a new kind of composition of automata and to show that this composition enjoys a lot of nice properties which do not hold for the direct product. We now give the definition of an automaton as used in this paper. Definition 1. An automaton A is a triple A = ( S , I , M ) where S is the state-set, I is the input-set and M : S X 1---~S is the transition function.We always assume that M also denotes the usual extension of the transition function M to S X I*.Before studying the cartesian composition we want to review shortly what is and is not known about the homogeneous and the heterogeneous direct product. For completeness we give the definitions of these concepts.

This article is a commentary on the papers in this issue of ZDM on Understanding Generalization in K-12 Algebra. It tries to interpret and understand at least some of the reported phenomena within a constructivist framework. The latter locates meaning not in the external representations but in the individual's activity on and with them. This activity on the other hand is strongly regulated (but not determined) by social contracts and belief systems. From that, and considerations of the more general mathematical context, various suggestions for further and extended research can be drawn. One special aspect is that generalization processes will have to be complemented by some kind of instruction on the conventional algebraic symbolism and its usage.

No abstract