R. Padmanabhan

Single identities for lattice theory and for weakly associative lattices

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 one-based, and we present a generalized one-based theorem for subvarieties of weakly associative lattices that can be defined with absorption laws. The automated theorem-proving program Otter was used in a substantial way to obtain the results.

Equational theory of algebras with a majority polynomial

This note gives a simple proof of a result of R. N. McKenzie that any finitely based variety of algebras which admit a majority polynomial and is definable by absorption identities is one-based. In the case of lattices, for example, this yields a “short” identity with only seven variables.

On Single Equational-Axiom Systems for Abelian Groups

It is a fascinating problem in the axiomatics of any mathematical system to reduce the number of axioms, the number of variables used in each axiom, the length of the various identities, the number of concepts involved in the system etc. to a minimum. In other words, one is interested finding systems which are apparently ‘of different structures’ but which represent the same reality. Sheffers stroke operation and. Byrnes brief formulations of Boolean algebras [1], Sholanders characterization of distributive lattices [7] and Sorkins famous problem of characterizing lattices by means of two identities are all in the same spirit. In groups, when defined as usual, we demand a binary, unary and a nullary operation respectively, say, a, b →a·b; a→a −1 ; the existence of a unit element). However, as G. Rabinow first proved in [6], groups can be made as a subvariety of groupoids (mathematical systems with just one binary operation) with the operation * where a * b is the right division, ab −1 . [8], M. Sholander proved the striking result that a mathematical system closed under a binary operation * and satisfying the identity S: x * ((x *z) * (y *z)) = y is an abelian group. Yet another identity, already known in the literature, characterizing abelian groups is HN: x * ((z * y) * (z * a;)) = y which is due to G. Higman and B. H. Neumann ([3], [4])*. As can be seen both the identities are of length six and both of them belong to the same ‘bracketting scheme’ or ‘bracket type’.

Two identities for lattices

0. Introduction. In this paper it is shown that every equational class of lattices which can be defined by a finite number of identities can be characterized (among the class of algebras with two binary operations) by means of two identities. In particular, the class of all lattices can be characterized by means of two identities and this solves a problem raised in [11] by Ju. I. Sorkin. In the proof of the main theorem we make use of the known fact that in presence of the lattice axioms the validity of a finite set of lattice identities is equivalent to that of a single one.

A single identity for Boolean groups and Boolean rings

Abstract By a well-known result of Higman and Neumann, Boolean groups (i.e., groups of exponent 2) can be represented as the class of groupoids which satisfy a single identity. In this paper we find all minimal length single identities which characterize Boolean groups and represent Boolean rings (associative rings with unit satisfying x2 = x) as the class of algebras with three operations and satisfying a single identity.

Placement of the Desargues configuration on a cubic curve

Hilbert and Cohn-Vossen once declared that the configurations of Desargues and Pappus are by far the most important projective configurations. These two are very similar in many respects: both are regular and self-dual, both could be constructed with ruler alone and hence exist over the rational plane, the final collinearity in both instances are ‘automatic’ and both could be regarded as self-inscribed and self-circumscribed p9lygons (see [1, p. 128]). Nevertheless, there is one fundamental difference between these two configurations, viz. while the Pappus-Brianchon configuration can be realized as nine points on a non-singular cubic curve over the complex plane (in doubly infinite ways), it is impossible to get such a representation for the Desargues configuration. In fact, the configuration of Desargues can be placed in a projective plane in such a way that its vertices lie on a cubic curve over a field k if and only if k is of characteristic 2 and has at least 16 elements. Moreover, any cubic curve containing the vertices of this configuration must be singular.

Otter and MACE

Otter 6] is a program that searches for proofs, and MACE 5] is a program that searches for small nite counterexamples. Both apply to statements in rst-order logic with equality. Otter is more powerful and exible at its task than MACE is at its task (consequently, Otter is more diicult to use). When searching for a proof with Otter the user typically formulates a search strategy and makes several attempts, modifying the strategy along the way. When searching for a counterexample with MACE, the user simply supplies a statement of the conjecture; if the conjecture is not too complex, and if there exist small nite models, then MACE will nd them. For many of our conjectures (quasigroup problems in particular), we know that counterexamples, if they exist, must be innnite; MACE is useless in such cases. But the two programs nicely complement one another in many other cases. Version 3.0.?? of Otter and version 1.0.?? of MACE were used for the experiments presented in this document. Both programs are in the public domain and are available by anonymous FTP. (All of the input les are also available by FTP.) See ftp://info.mcs.anl.gov/pub/Otter/README for information on obtaining the programs. The primary documentation for the programs are 6] and 5]; these are included with the programs when obtained by FTP. The following descriptions of the two programs are informal. See the manuals and 13] for more formal and detailed presentations. First, we deene some terms of automated theorem proving. 2.1 Deenitions These deenitions are biased toward Otter and equational theorem proving. A term is either a variable, a constant, or the application of an n-ary function symbol to n terms. An atom is the application of an n-ary predicate symbol to n terms. Nearly all atoms in this work will be equalities.