Results 1 
6 of
6
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
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.
Computer and Human Reasoning: Single Implicative Axioms for Groups and for Abelian Groups
, 1996
"... single axiom, but then \Delta is not product, and \Gamma1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and Supported ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
single axiom, but then \Delta is not product, and \Gamma1 is not inverse. The same situation holds for the Abelian case. Another curious fact is that there is no single equational axiom for groups or for Abelian groups in terms of the three standard operations of product, inverse, and Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W31109Eng38. the identity element [8]. Single equational axioms in terms of product and inverse have been reported by Neumann [5] and others [3, 2]. In this note we consider single implicative axioms, that is, axioms of the form ff = fi ) fl = ffi. For Abelian groups, an axiom of this type with five variables was given by Sholander [6]. If we allow one of f
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
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
Single Implicative Axioms for Groups and for Abelian Groups
, 1996
"... ) Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W31109Eng38. for abelian groups [3], and ((z \Delta (x \Delta y) \Gamma1 ) \Gamma1 \Delta (z \Delta ..."
Abstract
 Add to MetaCart
) Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W31109Eng38. for abelian groups [3], and ((z \Delta (x \Delta y) \Gamma1 ) \Gamma1 \Delta (z \Delta y \Gamma1 )) \Delta (y \Gamma1 \Delta y) \Gamma1 = x (4) for ordinary groups [2]. One might think it trivial, given (2), to obtain a single axiom in terms of product and inverse, by simply rewriting ff=fi to ff \Delta fi \Gamma1 . Doing so gives a single axiom, but then \Delta is not product, and \Gamma1
SINGLE AXIOMS: WITH AND WITHOUT COMPUTERS
"... This note is an (incomplete) summary of results on single equational axioms for algebraic theories. Pioneering results were obtained decades ago (without the use of computers) by logicians such asTarski, Higman, Neumann, and Padmanabhan. Use of today's highspeed computers and sophisticated software ..."
Abstract
 Add to MetaCart
This note is an (incomplete) summary of results on single equational axioms for algebraic theories. Pioneering results were obtained decades ago (without the use of computers) by logicians such asTarski, Higman, Neumann, and Padmanabhan. Use of today's highspeed computers and sophisticated software