Results 1 
9 of
9
Single Identities for Lattice Theory and for Weakly Associative Lattices
 Algebra Universalis
, 1995
"... . We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is onebased, and we present a generalized onebased theorem for subvarieties of weakly associative lattices that can be ..."
Abstract

Cited by 11 (10 self)
 Add to MetaCart
(Show Context)
. We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is onebased, and we present a generalized onebased theorem for subvarieties of weakly associative lattices that can be defined with absorption laws. The automated theoremproving program Otter was used in a substantial way to obtain the results. 1 Introduction Equational identities are, perhaps, the simplest form of sentences expressing many basic properties of algebras. Several familiar classes of algebras, such as semigroups, groups, rings, lattices, and Boolean algebras, are defined by equational identities. Such a class of algebras is known as an equational Supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W31109Eng38. y Supported by an operating grant from NSERC of Canada (#A8215). class of algebras or a variety of algebras (for mathematical properti...
Automated discovery of single axioms for ortholattices
 Algebra Universalis
, 2005
"... Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complemen ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complementation. Proofs are omitted but are available in an associated technical report and on the Web. We used computers extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. 1.
Short equational bases for ortholattices
 Preprint ANL/MCSP10870903, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 2004
"... Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. Proofs are omitted but are available in an associated technical report. Computers were used extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. The notion of computer proof is addressed. 1
Yet another single law for lattices
 Algebra Universalis
"... Abstract. In this note we show that the equational theory of all lattices is defined by the single absorption law (((y∨x)∧x)∨(((z∧(x∨x))∨(u∧x))∧v))∧(w∨((s∨x)∧(x∨t))) = x. This identity of length 29 with 8 variables is shorter than previously known such equations defining lattices. 1. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In this note we show that the equational theory of all lattices is defined by the single absorption law (((y∨x)∧x)∨(((z∧(x∨x))∨(u∧x))∧v))∧(w∨((s∨x)∧(x∨t))) = x. This identity of length 29 with 8 variables is shorter than previously known such equations defining lattices. 1.
Automated Equational Deduction with Otter
, 1995
"... Contents 1 Introduction 1 2 Otter and MACE 3 2.1 Otter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.1.1 Notes on Otter Proof Notation : : : : : : : : : : : : : : : 3 2.2 MACE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Test Chapter 3 3 Lattices a ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Contents 1 Introduction 1 2 Otter and MACE 3 2.1 Otter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.1.1 Notes on Otter Proof Notation : : : : : : : : : : : : : : : 3 2.2 MACE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Test Chapter 3 3 Lattices and Latticelike Structures 9 4 The Rule (gL) 23 4.1 Problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 4.2 Sample Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 5 Quasigroups 51 6 Semigroups 57 6.1 A Conjecture of Padmanabhan : : : : : : : : : : : : : : : : : : : 57 7 Groups 69 7.1 SelfDual Bases for Group Theory : : : : : : : : : : : : : : : : : 69 8 TC and RC 73 9 Problems not yet placed in the proper chapter 83 iii iv CONTENTS List
U.S.A.
, 1995
"... Abstract. We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is onebased, and we present a generalized onebased theorem for subvarieties of weakly associative lattices tha ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We present a single identity for the variety of all lattices that is much simpler than those previously known to us. We also show that the variety of weakly associative lattices is onebased, and we present a generalized onebased theorem for subvarieties of weakly associative lattices that can be dened with absorption laws. The automated theoremproving program Otter was used in a substantial way to obtain the results. 1
ANL/MCSTM265 Short Equational Bases for Ortholattices: Proofs and Countermodels
, 2003
"... Contract W31109ENG38. Argonne National Laboratory, with facilities in the states of Illinois and Idaho, is owned by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy. DISCLAIMER This report was prepared as an a ..."
Abstract
 Add to MetaCart
Contract W31109ENG38. Argonne National Laboratory, with facilities in the states of Illinois and Idaho, is owned by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor The University of Chicago, nor any of their employees or officers, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of document authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, Argonne National Laboratory, or The University of Chicago. ii