Results 1  10
of
14
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 ..."
Abstract

Cited by 48 (6 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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 =
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 ..."
Abstract
 Add to MetaCart
(Show Context)
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.
~) 1996 Birkh/iuser Verlag, Basel Undecidable
"... fragments of elementary theories ..."
(Show Context)