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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.).
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