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Normal Monomodal Logics Can Simulate All Others
 Journal of Symbolic Logic
, 1999
"... This paper shows that nonnormal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new ..."
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This paper shows that nonnormal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic. Normal monomodal logics can simulate all others 1 This paper is dedicated to our teacher, Wolfgang Rautenberg x1. Introduction. A simulation of a logic by a logic \Theta is a translation of the expressions of the language for into the language of \Theta such that the consequence relation defined by is reflected under the translation by the consequence relation of \Theta. A wellknown case is provided by the Godel translation, which simulates intuitionistic logic by Grzegorczyk's logic (cf. [11] and [5]). Such simulations not only yield technical results but may also ...
Equations in Algebra and Topology
, 1997
"... of the talk: In the early 1930's, Garrett Birkhoff introduced the notion a variety V , i.e. the class of all algebras (in the general sense) that model a fixed set \Sigma of equations, and proved his famous theorem characterizing such equational classes being classes closed under the formation o ..."
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of the talk: In the early 1930's, Garrett Birkhoff introduced the notion a variety V , i.e. the class of all algebras (in the general sense) that model a fixed set \Sigma of equations, and proved his famous theorem characterizing such equational classes being classes closed under the formation of subalgebras, homomorphic images and products. Ever since then (and more intensely since 1970), such classes have been actively studied  both in general and in many particular examples. In this talk we will survey some of this work, mentioning important results and problems of Tarski, Jonsson, McKenzie, Baker. We will also include some recent material on the modeling of the equations \Sigma by continuous operations on a topological space A. (Birkhoff was interested in this question as well.) Some recent results of the author say that many simple spaces A, e.g. a 2sphere or surface of genus 2, cannot continuously model any except the most trivial of equations \Sigma. 0.1 Garr...
Directions in Lattice Theory
, 1994
"... This paper is about three problems raised by Bjarni Jonsson and their influence on the development and direction of lattice theory ..."
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This paper is about three problems raised by Bjarni Jonsson and their influence on the development and direction of lattice theory
7. Varieties of Lattices
"... Variety is the spice of life. A lattice equation is an expression p ≈ q where p and q are lattice terms. Our intuitive notion of what it means for a lattice L to satisfy p ≈ q is that p(x1,...,xn) = q(x1,...,xn) whenever elements of L are substituted for the variables. This is captured by the forma ..."
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Variety is the spice of life. A lattice equation is an expression p ≈ q where p and q are lattice terms. Our intuitive notion of what it means for a lattice L to satisfy p ≈ q is that p(x1,...,xn) = q(x1,...,xn) whenever elements of L are substituted for the variables. This is captured by the formal definition: L satisfies p ≈ q if h(p) = h(q) for every homomorphism h: W(X) → L. We say that L satisfies a set Σ of equations if L satisfies every equation in Σ. Likewise, a class K of lattices satisfies Σ if every lattice L ∈ K does so. As long as we are dealing entirely with lattices, there is no loss of generality in replacing p and q by the corresponding elements of FL(X), since if terms p and p ′ evaluate the same in FL(X), then they evaluate the same for every substitution in every lattice. In practice it is often more simple and natural to think of equations between elements in a free lattice, rather than the corresponding terms, as in Theorem 7.2 below. A variety (or equational class) of lattices is the class of all lattices satisfying some
Properties. (description)
"... edit Definition 1. A modular lattice is a lattice L = 〈L, ∨, ∧ 〉 that satisfies the modular identity: ((x ∧ z) ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z) Definition 2. A modular lattice is a lattice L = 〈L, ∨, ∧ 〉 that satisfies the modular law: x ≤ z = ⇒ (x ∨ y) ∧ z ≤ x ∨ (y ∧ z) Definition 3. A modular latt ..."
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edit Definition 1. A modular lattice is a lattice L = 〈L, ∨, ∧ 〉 that satisfies the modular identity: ((x ∧ z) ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z) Definition 2. A modular lattice is a lattice L = 〈L, ∨, ∧ 〉 that satisfies the modular law: x ≤ z = ⇒ (x ∨ y) ∧ z ≤ x ∨ (y ∧ z) Definition 3. A modular lattice is a lattice L = 〈L, ∨, ∧ 〉 such that L has no sublattice isomorphic to the pentagon N5 Morphisms. Let L and M be modular lattices. function h: L → M that is a homomorphism: h(x ∨ y) = h(x) ∨ h(y), h(x ∧ y) = h(x) ∧ h(y) Basic Results. Examples. A morphism from L to M is a 1. M3 is the smallest nondistributive modular lattice. By a result of [1] this lattice occurs as a sublattice of every nondistributive modular lattice.
7. Varieties of Lattices
"... Variety is the spice of life. A lattice equation is an expression p ≈ q where p and q are lattice terms. Our intuitive notion of what it means for a lattice L to satisfy p ≈ q is that p(x1,...,xn) = q(x1,...,xn) whenever elements of L are substituted for the variables. This is captured by the forma ..."
Abstract
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Variety is the spice of life. A lattice equation is an expression p ≈ q where p and q are lattice terms. Our intuitive notion of what it means for a lattice L to satisfy p ≈ q is that p(x1,...,xn) = q(x1,...,xn) whenever elements of L are substituted for the variables. This is captured by the formal definition: L satisfies p ≈ q if h(p) = h(q) for every homomorphism h: W(X) → L. We say that L satisfies a set Σ of equations if L satisfies every equation in Σ. Likewise, a class K of lattices satisfies Σ if every lattice L ∈ K does so. As long as we are dealing entirely with lattices, there is no loss of generality in replacing p and q by the corresponding elements of FL(X), since if terms p and p ′ evaluate the same in FL(X), then they evaluate the same for every substitution in every lattice. In practice it is often more simple and natural to think of equations between elements in a free lattice, rather than the corresponding terms, as in Theorem 7.2 below. A variety (or equational class) of lattices is the class of all lattices satisfying some