Matrix Expression of Finite Boolean-type Algebras
Daizhan Cheng, Jun-e Feng, Jianli Zhao, Shihua Fu
TL;DR
This paper explores finite Boolean-type algebras by decomposing them into lattice and complementation components, providing matrix representations, and establishing conditions for their decomposition and universal generation.
Contribution
It introduces matrix expressions for finite Boolean-type lattices and complementation algebras, and presents a universal generator for finite universal algebras.
Findings
Matrix expressions for finite BTL and CA are derived.
Conditions for BTA decomposition are established.
A universal generator for finite universal algebras is proposed.
Abstract
Boolean-type algebra (BTA) is investigated. A BTA is decomposed into Boolean-type lattice (BTL) and a complementation algebra (CA). When the object set is finite, the matrix expressions of BTL and CA (and then BTA) are presented. The construction and certain properties of BTAs are investigated via their matrix expression, including the homomorphism and isomorphism, etc. Then the product/decomposition of BTLs are considered. A necessary and sufficient condition for decomposition of BTA is obtained. Finally, a universal generator is provided for arbitrary finite universal algebras.
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Taxonomy
TopicsAdvanced Algebra and Logic · Logic, Reasoning, and Knowledge · Rough Sets and Fuzzy Logic