# Primal clusters and local binary algebras

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## Abstract

Primal clusters and local binary algebras ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze ADILYAQUB Primal clusters and local binary algebras Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 21, no 2 (1967), p. 111-119. <http://www.numdam.org/item?id=ASNSP_1967_3_21_2_111_0> © Scuola Normale Superiore, Pisa, 1967, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ PRIMAL CLUSTERS AND LOCAL BINARY ALGEBRAS ADIL YAQUB The theory of a primal (= strictly functionally complete) algebra sub- sumes and substantially generalizes the classical Boolean theory as well as that of p-rings and Post algebras. Here a primal algebra is a finite algebra in which each map is expressible in terms of the primitive operations of the algebra. The concept of independence is essentially a generalization to universal algebras of the Chinese remainder theorem in number theory. A primal cluster is a class ( Ui) of universal algebras Ui of the same species in which each Ui is primal and such that every finite subset of (Uil is in- dependent. Our present object is to show that the class ((Bn , x)) of all « local » binary algebras of distinct orders, endowed with a suitably chosen permu- tation ~‘ of Bi, forms a primal cluster ~, -)I (of species (2, 1)). Here a local binary algebra is a finite associative binary algebra (B, x) such that every element in B is either nilpotent or has an inverse (in B). In Theorem 5, which is our ma

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