Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.

It is possible to define a general notion of cascade composition for tree-automata in a way very similar to the case of ordinary automata. For this cascade composition we show necessary and sufficient conditions of decomposability like those of the ordinary case. Also some kinds of associativity properties continue to hold.A further generalization leads us to apply these results to a large class of universal algebras, and by some examples it is shown that we can decompose some familiar algebras into cascades of tree-automata. Moreover, the cascades obtained identify some algorithms that are the common ones for computing in the algebras considered.1. Introduction. The cascade composition of semiautomata [4] (or state machines [6]) is one of the elementary notions of Structure Theory for ordinary automata. Yet no similar notion is used in Universal Algebra where another and simpler kind of composition (the direct product) plays the main role.The purpose of this paper is to extend the notion of cascade composition from Automata Theory to Universal Algebra. Indeed, such extension is possible and we will show also that the main algebraic properties of the cascade of semiautomata carry over to the general case (at least for a large class of universal algebras). These are the properties concerning decomposability by congruences or S-P partitions and associativity [4].Why do we have to consider this extension? There are two possible answers. One concerns a particular class of algebras, the tree-automata In the papers mentioned above many properties of ordinary automata have been shown to hold also for tree-automata. Moreover in [7] there is a first attempt (different from ours) to investigate the decomposition properties of a subclass of tree-automata by a particmar notion of cascade. Our outlined results add some other infoni~:,tion in that direction. (We point out that the

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Summary. --Detailed comparisons between Montecarlo and Boltzmann calculations of electron energy distributions in gases acted upon by RF fields are presented. Attention is turned to model gases of special theoretical interest but various calculations have also been made for real gases such as pure CO and He-CO mixtures. The analysis has shown that large discrepancies exist between energy distributions obtained with the two mentioned techniques under conditions of particular physical interest. The discrepancies are found to be the consequence of the two-term approximation and are expected to disappear if an appropriate multiterm solution of the Boltzmann equation is adopted.PACS 51.50 -Electrical phenomena in gases.

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