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Properties of the symmetric difference in lattices with complementation
Václav Cenker1, Ivan Chajda1, Helmut Länger2,1
1Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. Listopadu 12, CZ-771 46 Olomouc, Czech Republic.
Mathematica Slovaca
|June 29, 2026
Summary
The symmetric difference in Boolean lattices has two equivalent forms. This study explores lattices where these forms coincide, proving associativity and De Morgan
Area of Science:
- Lattice Theory
- Abstract Algebra
- Boolean Algebra
Background:
- The symmetric difference in Boolean lattices has two equivalent definitions.
- These definitions may not coincide in general bounded lattices with complementation.
Purpose of the Study:
- To investigate lattices with complementation where the two forms of symmetric difference coincide.
- To analyze the properties of associativity and specific identities related to symmetric difference.
- To characterize lattices with a unary operation satisfying De Morgan's laws.
Main Methods:
- Studying varieties of lattices with complementation.
- Utilizing a result by J. Berman to estimate the size of free algebras.
- Proving characterization theorems for Boolean lattices based on symmetric difference properties.
Main Results:
- Identified subvarieties of lattices where the two symmetric difference expressions are equivalent.
- Demonstrated that symmetric difference is associative in a lattice with complementation if and only if the lattice is Boolean.
- Established that a lattice with complementation is Boolean if and only if the symmetric difference satisfies a specific two-variable identity.
- Characterized lattices with a unary operation adhering to De Morgan's laws.
Conclusions:
- The associativity of symmetric difference is a defining characteristic of Boolean lattices.
- Specific identities involving symmetric difference provide alternative characterizations of Boolean lattices.
- The study contributes to the understanding of lattice structures and their algebraic properties.
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