Vol. 16, No. 2, 1966

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Primal clusters

Adil Mohamed Yaqub

Vol. 16 (1966), No. 2, 379–388

In a series of recent publications [Math. Z, 66 (1957), 452–469; Math Z, 62 (1955), 171–188] Foster introduced and studied the theory of a “primal cluster”, —a concept which embraces classes of algebras of such diverse nature as the classes of all (i) prime-fields, (ii) “n-fields”, (iii) basic Post algebras. Here, a primal cluster is essentially a class {Ui} of primal (= strictly functionally complete) algebras of the same species such that every finite subset of {Ui} is “independent”. The concept of independence is essentially a generalization to universal algebras of the Chinese residue Theorem in number theory. Each cluster, Ũ, equationally defines—in terms of the identities jointly satisfied by the various finite subset of Ũ-a class of “Ũ-algebras”, and a structure theory for these Ũ-algebras was established by Foster,—a theory which contains well known results for Boolean rings, p-rings, and Post algebras. In order to expand the domain of applications of this theory, one should then look for primal clusters. In this paper a permutation, , of ths residue class ring Rn, mod n, is constructed, such that {(Rn,×,)} forms a primal cluster. In Theorem 9, which is the main result of this paper, it is shown that a much more comprehensive (and quite “heterogeneous”) class K of algebras nevertheless forms a primal cluster. Indeed, K here is the union of all nonisomorphic algebras in the classes of all (i) residue class rings, (ii) basic Post algebras, and (iii) “n-fields”. Thus, the primal cluster K furnishes an extension of the primal clusters which were previously given by Foster (loc. cit.).

Mathematical Subject Classification
Primary: 08.10
Received: 28 July 1964
Revised: 12 October 1964
Published: 1 February 1966
Adil Mohamed Yaqub