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Representations by algebras of sections over Boolean spaces. (English) Zbl 0219.08002

A sheaf of universal algebras \((X,S)\) is called reduced if it is trivial or the following conditions hold: (i) \(X\) is a Boolean space; (ii) each stalk is nontrivial; (iii) the factor congruence relations on \(\Gamma(X,S)\), the algebra of all continuous sections of \((X,S)\), form a sublattice of \(\Theta(\Gamma(X,S))\) which is isomorphic to the BA of all clopen subsets of \(X\) in a natural way. The main result is as follows:
Suppose a universal algebra \(A\) satisfies the conditions (I) the set \(\Theta_0(A)\) of all factor congruence relations of \(A\) form a sublattice of \(\Theta(A)\) which is a BA under \(\vert\) and \(\cap\) and (II) the congruence relation generated by a proper BA ideal of \(\Theta_0(A)\) is proper. Then there exist a unique (up to sheaf isomorphism) reduced sheaf \((X(A),S(A))\) of algebras such that \(A\) is isomorphic to \(\Gamma(X(A),S(A))\).
The above result is an extension to universal algebras of the sheaf representation result for rings with \(1\) given by R. S. Pierce [Modules over commutative regular rings. Mem. Am. Math. Soc. 70, 112 p. (1967; Zbl 0152.02601)]. The result also applies to various other types of algebraic structures such as cylindric and polyadic algebras and lattices with \(0\) and \(1\).

MSC:

03G99 Algebraic logic
08A30 Subalgebras, congruence relations

Citations:

Zbl 0152.02601
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