Abstract not found

We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke than the ones we present. Automated deduction techniques were used for several different aspects of the work. Keywords: Boolean algebra, Sheffer stroke, single axiom 1. Background and

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 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

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 Lattice-like 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 Self-Dual Bases for Group Theory : : : : : : : : : : : : : : : : : 69 8 TC and RC 73 9 Problems not yet placed in the proper chapter 83 iii iv CONTENTS List

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