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Prime ideal theory for general algebras. (English) Zbl 0980.08001
We may be given collections \(\mathcal A\) and \(\mathcal B\) of subsets of a ground set \(G\) (endowed with a relational and/or a topological structure). Given any two disjoint sets \(A\in \mathcal A\) and \(B\in \mathcal B\) we want to extend \(A\) to a member of \(\mathcal A\) whose complement belongs to \(\mathcal B\) and contains \(B\). If this is always possible, the members of \(\mathcal A\) are called totally separated from those of \(\mathcal B\). Statements of this kind are called separation lemmas. We assume that \(\mathcal A\) is a set-theoretical closure system. A subset of \(G\) is called prime if its complement belongs to \(\mathcal B\), and semiprime if its complement is a union of sets in \(\mathcal B\). In algebraic contexts where \(\mathcal A\) consists of certain “ideals”, separation lemmas or the corresponding intersection theorems are also called prime ideal theorems.
The author introduces ideals, radicals and prime ideals in arbitrary algebras with at least one binary operation and shows that various separation lemmas and prime ideal theorems are special instances of one general theorem that is equivalent to the Boolean Prime Ideal Theorem (or the Ultrafilter Principle).

08A30 Subalgebras, congruence relations
06F05 Ordered semigroups and monoids
20M12 Ideal theory for semigroups
20N02 Sets with a single binary operation (groupoids)
03G05 Logical aspects of Boolean algebras
16D30 Infinite-dimensional simple rings (except as in 16Kxx)
16N80 General radicals and associative rings
17A65 Radical theory (nonassociative rings and algebras)
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