# Short Equational Bases for Ortholattices: Web Support

William McCune, R. Padmanabhan, Michael A. Rose, Robert Veroff

There are several parts to the documentation of the project.

• A short paper for a journal.
• Version v16: PDF, tex (Sept 10, 2004), minor revision from v15, as suggested by referee.
• Version v15 (Feb 23, 2004), submitted for publication.
• A paper for a wide audience.
• Version v10 (Oct 30, 2003); The American Mathematical Monthly declined to publish this.
• A set of Web pages (this page is the root) containing proofs, countermodels, and additional background and explanation. These are for immediate support of the papers.
• A technical report containing proofs and countermodels. This is for archival support of the papers.

Son of BirdBrain, the Web-based demo of Otter and Mace2, can prove most of the easy theorems and find most of the countermodels cited in the papers. Try it now --- select the area "Ortholattices".

## Abstract (from the v10 paper)

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 Introduction

The structure of this page roughly follows the v10 paper, with similar section titles.

The following notes give some explanation of the proofs and countermodels.

## 2 Equational Bases

### 2.1 In Terms of Join/Meet/Complement

These links lead to the join/meet/complement bases for the varieties, including proofs of various properties and independence proofs cited in the papers.

### 2.2 In Terms of the Sheffer Stroke

These links lead to the Sheffer stroke bases for the varieties. Proofs that the bases are definitionally equivalent to the join/meet/complement bases are given, and independence proofs are given.

Our Sheffer stroke notation is inconsistent --- for Otter and Mace we use "f(x,y)", and when formatting the equations for presentation we use "(x | y)".

### 2.3 Finding and Proving the Multiequation Bases

See the v10 paper.

### 2.4 Are There Simpler Multiequation Bases?

See the v10 paper.

## 3 Single Axioms

### 3.1 Generating and Filtering Candidates

See "Single Axiom Searches and Proofs" below.

### 3.2 Finite Ortholattices

The table Number of Lattices gives access to listings of the small finite OLs, OMLs, MOLs, and BAs.

### 3.3 Collecting and Applying Filters

How were the filters found?

### 3.4 Trying to Prove that Candidates are Single Axioms

See "Single Axiom Searches and Proofs" below.

### 3.5 Single Axioms for OL, OML, and MOL

```    f(f(f(f(y,x),f(x,z)),u),f(x,f(f(x,f(f(y,y),y)),z))) = x.       % OL-Sh
f(f(f(f(y,x),f(x,z)),u),f(x,f(f(z,f(f(x,x),z)),z))) = x.       % OML-Sh
f(f(y,x),f(f(f(x,x),z),f(f(f(f(f(x,y),z),z),x),f(x,u)))) = x.  % MOL-Sh
```
For comparison, here is one of the BA axioms .
```    f(f(y,f(f(x,y),y)),f(x,f(z,y))) = x.   % BA-Sh
```

## 4 The Computer Programs

Here is a summary of the programs, with examples. Also see the v10 paper.

## 5 Remarks

See the v10 paper.

## References

These references are taken from the v10 paper and the tech. report.
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3. D. Kelly and R. Padmanabhan. Orthomodular lattices and congruence permutability. Preprint, 2003.
4. W. McCune. MACE 2.0 Reference Manual and Guide. Tech. Memo ANL/MCS-TM-249, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, June 2001.
5. W. McCune. Mace4 Reference Manual and Guide. Tech. Memo ANL/MCS-TM-263, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, August 2003.
6. W. McCune. Otter 3.3 Reference Manual. Tech. Memo ANL/MCS-TM-263, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, August 2003.
7. W. McCune and R. Padmanabhan. Single identities for lattice theory and for weakly associative lattices. Algebra Universalis, 36(4):436--449, 1996.
8. W. McCune, R. Padmanabhan, M. A. Rose, and R. Veroff. Short equational bases for ortholattices: Proofs and countermodels. Tech. Memo ANL/MCS-TM-265, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, September 2003.
9. W. McCune, R. Padmanabhan, M. A. Rose, and R. Veroff. Short equational bases for ortholattices. Preprint ANL/MCS-P1087-0903, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, September 2003.
10. W. McCune, R. Padmanabhan, and R. Veroff. Yet another single law for lattices. Algebra Universalis. To appear.
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20. R. Veroff. A shortest 2-basis for Boolean algebra in terms of the Sheffer stroke. J. Automated Reasoning. To appear.
21. R. Veroff. Using hints to increase the effectiveness of an automated reasoning program: Case studies. J. Automated Reasoning, 16(3):223--239, 1996.
22. R. Veroff. Solving open questions and other challenge problems using proof sketches. J. Automated Reasoning, 27(2):157--174, 2001.