ISSN:
1420-8911
Keywords:
action of a Boolean ring on a set, sheaf of sets on the spectrum of a Boolean ring, commuting rectangular band operations
;
bounded Boolean power of a set or algebra
;
least nontrivial hypervariety of algebras
;
Primary: 06E15
;
06E20
;
18F20
;
secondary: 08A05
;
08B25
;
18C10
;
20M50
;
54H10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract LetB be a Boolean ring (with 1),S a sheaf of sets on the Stone space Spec(B), andS the set of global sections of S. For everya εB ands, t εS, leta(s, t) denote the element ofS which agrees withs on the support ofa, and witht elsewhere. We set down identities satisfied by this ternary operationB×S×S→S (involving also the Boolean operations ofB). For a fixed Boolean ringB, we call a setS given with a ternary operation satisfying these identities aBset. The above construction is shown to give a functorial equivalence between sheaves of setsS on Spec(B) with nonempty sets of global sections, and nonemptyB-setsS. For any setA, the bounded Boolean powerA[B]* is the freeB-set onA. The varieties ofB-sets, asB ranges over all Boolean rings, constitute (together with one trivial variety) the least nontrivial hypervariety of algebras, in the sense of W. Taylor.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01190851
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