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Congruences and Boolean filters of quasi-modular p-algebras

100%
El-Mohsen Badawy A., Shum K.
Discussiones Mathematicae - General Algebra and Applications
|
2014
|
tom 34
|
nr 1
109-123
EN
The concept of Boolean filters in p-algebras is introduced. Some properties of Boolean filters are studied. It is proved that the class of all Boolean filters BF(L) of a quasi-modular p-algebra L is a bounded distributive lattice. The Glivenko congruence Φ on a p-algebra L is defined by (x,y) ∈ Φ iff x** = y**. Boolean filters [Fₐ), a ∈ B(L) , generated by the Glivenko congruence classes Fₐ (where Fₐ is the congruence class [a]Φ) are described in a quasi-modular p-algebra L. We observe that the set $F_{B}(L) = {[Fₐ): a ∈ B(L)}$ is a Boolean algebra on its own. A one-one correspondence between the Boolean filters of a quasi-modular p-algebra L and the congruences in [Φ,∇] is established. Also some properties of congruences induced by the Boolean filters [Fₐ), a ∈ B(L) are derived. Finally, we consider some properties of congruences with respect to the direct products of Boolean filters.
2
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Minimal formations of universal algebras

100%
Guo W., Shum K.
Discussiones Mathematicae - General Algebra and Applications
|
2001
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tom 21
|
nr 2
201-205
EN
A class ℱ of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A ∈ ℱ is in ℱ; 2) If α₁, α₂ are congruences on A and $A/α_{i} ∈ ℱ$, i = 1,2, then A/(α₁∩α₂) ∈ ℱ. We prove that any formation generated by a simple algebra with permutable congruences is minimal, and hence any formation containing a simple algebra, with permutable congruences, contains a minimum subformation. This result gives a partial answer to an open problem of Shemetkov and Skiba on formations of finite universal algebras proposed in 1989.
3
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Vague ideals of implication groupoids

100%
Bandaru R., Shum K.
Discussiones Mathematicae - General Algebra and Applications
|
2013
|
tom 33
|
nr 2
221-231
EN
We introduce the concept of vague ideals in a distributive implication groupoid and investigate their properties. The vague ideals of a distributive implication groupoid are also characterized.
4
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Sturdy frames of type (2,2) algebras and their applications to semirings

81%
Zhao X., Guo Y., Shum K.
Fundamenta Mathematicae
|
2003
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tom 179
|
nr 1
69-81
EN
We introduce sturdy frames of type (2,2) algebras, which are a common generalization of sturdy semilattices of semigroups and of distributive lattices of rings in the theory of semirings. By using sturdy frames, we are able to characterize some semirings. In particular, some results on semirings recently obtained by Bandelt, Petrich and Ghosh can be extended and generalized.
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