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Splittings of lattices of theories and unification types. (English) Zbl 1109.06008
Dorfer, G. (ed.) et al., Proceedings of the 70th workshop on general algebra “70. Arbeitstagung Allgemeine Algebra”, Vienna, Austria, May 26–29, 2005. Klagenfurt: Verlag Johannes Heyn (ISBN 3-7084-0194-8/pbk). Contributions to General Algebra 17, 71-81 (2006).
Summary: We show that the lattice of all theories extending the equational theory of Heyting algebras is split into two parts, “upper” and “lower”. Such a splitting is related to unification types: the upper part contains all theories having unitary unification type (and some of nullary type), the lower part contains all theories having finitary unification type (and some of nullary type, plus some with infinitary type, if there are such). The same splitting determines a limit between theories of constructive (upper part) and non-constructive (lower part) character. Similar results are proved for the lattice of all theories extending the equational theory of interior algebras (or topological Boolean algebras).
For the entire collection see [Zbl 1089.08001].

MSC:
06D20 Heyting algebras (lattice-theoretic aspects)
03B35 Mechanization of proofs and logical operations
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
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