Howlader, Prosenjit

Kwuida, Léonard

Behrisch, Mike

Liau, Churn-Jung

2025-07-22T09:06:29Z

2025-07-22T09:06:29Z

2025-06-22

<div class="csl-bib-body"> <div class="csl-entry">Howlader, P., Kwuida, L., Behrisch, M., &#38; Liau, C.-J. (2025, June 22). <i>Independent set of axioms of double Boolean algebras</i> [Conference Presentation]. AAA 107 - 107th Workshop on General Algebra, Bern, Switzerland. http://hdl.handle.net/20.500.12708/217195</div> </div>

http://hdl.handle.net/20.500.12708/217195

Due to indisposition of Prosenjit Howlader, the conference talk was given by Mike Behrisch.

The notion of double Boolean algebra was introduced by Wille to explore the equational class of protoconcept algebras. Many authors have studied fundamental results, including the Prime Ideal Theorem, topological representation, and the congruence structure for dBas. Double Boolean algebras are defined by a set of 23 equations given in the subsequent definition. Definition: An algebra 𝕯 := (D, ⊔, ⊓, ¬, ⌟, ⊤, ⊥) satisfying the following properties is called a double Boolean algebra (dBa). For any 𝑥, 𝑦, 𝑧 ∈ D we require (1a)(𝑥 ⊓ 𝑥) ⊓ 𝑦 = 𝑥 ⊓ 𝑦 (1b)(𝑥 ⊔ 𝑥) ⊔ 𝑦 = 𝑥 ⊔ 𝑦 (2a)𝑥 ⊓ 𝑦 = 𝑦 ⊓ 𝑥 (2b)𝑥 ⊔ 𝑦 = 𝑦 ⊔ 𝑥 (3a)¬(𝑥 ⊓ 𝑥) = ¬𝑥 (3b)⌟(𝑥 ⊔ 𝑥) = ⌟𝑥 (4a)𝑥 ⊓ (𝑥 ⊔ 𝑦) = 𝑥 ⊓ 𝑥 (4b)𝑥 ⊔ (𝑥 ⊓ 𝑦) = 𝑥 ⊔ 𝑥 (5a)𝑥 ⊓ (𝑦 ∨ 𝑧) = (𝑥 ⊓ 𝑦) ∨ (𝑥 ⊓ 𝑧) (5b)𝑥 ⊔ (𝑦 ∧ 𝑧) = (𝑥 ⊔ 𝑦) ∧ (𝑥 ⊔ 𝑧) (6a)𝑥 ⊓ (𝑥 ∨ 𝑦) = 𝑥 ⊓ 𝑥 (6b)𝑥 ⊔ (𝑥 ∧ 𝑦) = 𝑥 ⊔ 𝑥 (7a)¬¬(𝑥 ⊓ 𝑦) = 𝑥 ⊓ 𝑦 (7b)⌟⌟(𝑥 ⊔ 𝑦) = 𝑥 ⊔ 𝑦 (8a)𝑥 ⊓ ¬𝑥 = ⊥ (8b)𝑥 ⊔ ⌟𝑥 = ⊤ (9a)¬⊤ = ⊥ (9b)⌟⊥ = ⊤ (10a)𝑥 ⊓ (𝑦 ⊓ 𝑧) = (𝑥 ⊓ 𝑦) ⊓ 𝑧 (10b)𝑥 ⊔ (𝑦 ⊔ 𝑧) = (𝑥 ⊔ 𝑦) ⊔ 𝑧 (11a)¬⊥ = ⊤ ⊓ ⊤ (11b)⌟⊤ = ⊥ ⊔ ⊥ (12)(𝑥 ⊓ 𝑥) ⊔ (𝑥 ⊓ 𝑥) = (𝑥 ⊔ 𝑥) ⊓ (𝑥 ⊔ 𝑥) where 𝑥 ∨ 𝑦 ≔ ¬(¬𝑥 ⊓ ¬𝑦) and 𝑥 ∧ 𝑦 ≔ ⌟(⌟𝑥 ⊔ ⌟𝑦). This is a relatively long list, which might refrain potential users to engage in investigating or applying these structures. In previous work, it was shown that the axioms (1a), (1b) and (11a), (11b) can be derived from other axioms of dBa. During a conference in Cádiz, the authors identified a shared interest in simplifying the axiomatization of dBas. As a result, we collaborated on developing a more streamlined, redundancy-free axiomatization. Let DAxom be the set of all axioms of dBa and MCore be the subset of DAxom consisting of the axioms 2a, 2b, 3a, 3b, 4a, 4b, 5a, 5b 8a, 8b and 12. Then we have the following result: Theorem: There is a six-element non-dBa 𝕯 such that 𝕯 satisfies all the equations in MCore but does not satisfy the axioms 7a and 7b. The above theorem suggests that we need MCore and the axioms 7a and 7b to define a dBa. We denote DCore := MCore ∪ {¬¬(x ⊓ y) = x ⊓ y, ⌟⌟(x ⊔ y) = x ⊔ y}. Theorem: An algebra (D, ⊔, ⊓, ¬, ⌟, ⊤, ⊥) is a dBa if and only if it satisfies DCore. Moreover DCore is a minimal set of axioms for such an algebra.

en

Formal Concept Analysis

double Boolean algebra

Boolean algebra

Independent set of axioms of double Boolean algebras

Presentation

Vortrag

Institute of Information Science, Academia Sinica, Taiwan (Province of China)

Bern University of Applied Sciences, Switzerland

Institute of Information Science, Academia Sinica, Taiwan (Province of China)

Conference Presentation

I1

A3

Logic and Computation

Fundamental Mathematics Research

25

75

E104-01 - Forschungsbereich Algebra

E104 - Institut für Diskrete Mathematik und Geometrie

0000-0002-8932-126X

0000-0002-9811-0747

0000-0003-0050-8085

0000-0001-6842-9637

AAA 107 - 107th Workshop on General Algebra

Swiss National Science Foundation (SNSF)

IZSEZ0_233403 / 1

20-06-2025

22-06-2025

On Site

Event for scientific audience

Bern

CH

Bern University of Applied Sciences (BFH-W)

Behrisch, Mike

Single Track

Informatik

Mathematik

1020

1010

25

75

en

conference paper not in proceedings

http://purl.org/coar/resource_type/c_18cp

none

Publications

no Fulltext

Institute of Information Science, Academia Sinica

Bern University of Applied Sciences

E104-01 - Forschungsbereich Algebra

Institute of Information Science, Academia Sinica

0000-0002-8932-126X

0000-0003-0050-8085

0000-0001-6842-9637

E104 - Institut für Diskrete Mathematik und Geometrie