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Relational quantifiers

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Krynicki M.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction.................................................................................5 1. Algebras of monotone quantifiers.............................................7 1.1. Family of monotone quantifiers............................................7 1.2. Lattice of monotone quantifiers............................................8 1.3. Other operations in M(κ)......................................................9 2. Algebras of relational quantifiers............................................11 2.1. Basic properties of the family of relational quantifiers........11 2.2. Relations and quantifiers...................................................13 2.3. Lattices of relational quantifiers.........................................15 2.4. Other operations in R(κ)....................................................17 3. Logics with relational quantifiers............................................18 3.1. Structures with relational quantifiers..................................18 3.2. Completeness theorem......................................................19 3.3. Some simple consequences...............................................20 4. Model theory of relational quantifiers.....................................21 4.1. Basic notions......................................................................21 4.2. Substructure relations and preservation therems..............23 4.3. The chain property.............................................................27 4.4. Product operations............................................................29 5. Classes of relations and their logics......................................33 5.1. Logics determined by classes of relations.........................33 5.2. Classes of relations vs. sets of sentences.........................36 5.3. The Galois connection.......................................................38 5.4. The lattice of closed classes..............................................40 5.5. Further properties..............................................................44 5.6. Some open questions........................................................45 References...............................................................................46

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1 Krynicki M.

1 1995

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Relational quantifiers