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A Survey of Residuated Lattices
"... Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of latticeordered ..."
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Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of latticeordered groups, ideal lattices of rings, linear logic and multivalued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability. We end with a list of open problems that we hope will stimulate further research.
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 ...
On modular lattices generated by two complemented pairs
 Houston J. Math
"... In [1; Problem 43] G. Birkhoff called for a description of the modular lattice FM(B) freely generated by four elements a, b, c, and d satisfying a+d = b+c = I and ad = bc = 0. Examples of subdirectly irreducible factors (as it turns out all of them) are provided in the paper [2] of A. Day, R. Wille ..."
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In [1; Problem 43] G. Birkhoff called for a description of the modular lattice FM(B) freely generated by four elements a, b, c, and d satisfying a+d = b+c = I and ad = bc = 0. Examples of subdirectly irreducible factors (as it turns out all of them) are provided in the paper [2] of A. Day, R. Wille et al.. There the modular lattice 14 FM(J) freely generated by elements a,b,c,d subject to the relations ab = ac = ad =
Some order theoretic questions about free lattices and free modular lattices, ordered Sets
, 1982
"... In this paper we look at some of the problems on free lattices and free modular lattices which are of an order theoretic nature. We review some of the known results, give same new results, and present several open problems. Every countable partially ordered set can be order embedded into a countable ..."
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In this paper we look at some of the problems on free lattices and free modular lattices which are of an order theoretic nature. We review some of the known results, give same new results, and present several open problems. Every countable partially ordered set can be order embedded into a countable free lattice [6]. However, free lattices contain no uncountable chains [25],so the above result does not extend to arbitrary partially ordered sets. The problem of which partially ordered sets can be embedded into a free lattice is open. It is not enough to require that the partially ordered set does not have any uncountable chains. In fact, there are partially ordered sets of height one, which cannot be embedded into any free lattice [23].· The importance of these concepts to projective lattices is discussed. If a> b and there is no a with a> a> b we say a aoVerB b and write ar b. Covers in free lattices and free modular lattices are important to lattice structure theory. We discuss the connec~ion between covers and structure theory and give same of the inore important results about covers. Alan Day has shown that every quotient sublattice (i.e. interval) of a finitely generated free lattice contains a covering [9]. R.A. Dean, on the other hand, has some results on noncovers in free lattices. The analogous problems for free modular lattices are open. Suppose w(x,...,x) is a lattice word and L is a lattice. 1 n _ If we replace all but one of the variables with fixed elements from L we obtain a function f(x) = w(x,a2, •• •,an) fram L to L.
Full lambek calculus with contraction is undecidable
"... Among propositional substructural logics, these obtained from Gentzen’s sequent calculus for intuitionistic logic (LJ) by removing a subset of the rules ..."
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Among propositional substructural logics, these obtained from Gentzen’s sequent calculus for intuitionistic logic (LJ) by removing a subset of the rules
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.
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 format ..."
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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...