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Algebraic logic, varieties of algebras, and algebraic varieties
, 1995
"... Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could ..."
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Cited by 14 (5 self)
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Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.
Notions Of Density That Imply Representability In Algebraic Logic
, 1998
"... Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable (as a cylindric set algebra) . This theorem and its analogues for quasipolyadic algebras with and without equality are formulated in HenkinMonkTarski [1985]. We introduce a natur ..."
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Cited by 5 (1 self)
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Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable (as a cylindric set algebra) . This theorem and its analogues for quasipolyadic algebras with and without equality are formulated in HenkinMonkTarski [1985]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasipolyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result to other classes of algebras of logic, for example quasipolyadic algebras and substitutioncylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the HenkinMonkTarski representation theorem for atomic equalityfree quasipolyadic algebras with rectangular atoms. The er...