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This monograph is the result of the cooperation of a mathematician working in universal algebra and geometry, and a computer scientist working in automated deduction, who succeeded in employing the theorem prover Otter for proving first order theorems from mathematics and then intensified their joint effort.Mathematicians will find many new results from equational logic, universal algebra, and algebraic geometry and benefit from the state-of-the-art outline of the capabilities of automated deduction techniques. Computer scientists will find a large and varied source of theorems and problems that will be useful in designing and evaluation automated theorem proving systems and strategies.
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The importance of equational axioms emerged initially with the axiomatic approach to Boolean algebras, groups, and rings, and later in lattices. This unique research monograph systematically presents minimal equational axiom-systems for various lattice-related algebras, regardless of whether they are given in terms of “join and meet” or other types of operations such as ternary operations. Each of the axiom-systems is coded in a handy way so that it is easy to follow the natural connection among the various axioms and to understand how to combine them to form new axiom systems.A new topic in this book is the characterization of Boolean algebras within the class of all uniquely complemented lattices. Here, the celebrated problem of E V Huntington is addressed, which — according to G Gratzer, a leading expert in modern lattice theory — is one of the two problems that shaped a century of research in lattice theory. Among other things, it is shown that there are infinitely many non-modular lattice identities that force a uniquely complemented lattice to be Boolean, thus providing several new axiom systems for Boolean algebras within the class of all uniquely complemented lattices. Finally, a few related lines of research are sketched, in the form of appendices, including one by Dr Willian McCune of the University of New Mexico, on applications of modern theorem-proving to the equational theory of lattices.