• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1587-1606
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• 2015

Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic $J{\acute{o}}nsson$-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying $J{\acute{o}}nsson$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $J{\acute{o}}nsson$-Tarski monoid forms a group is open.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.457-463
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• 2007

We show that any filter of Tarski algebra can be de-composed into the union of some sets. Moreover, we introduce the notion of expansions of filters in Tarski algebras, and discuss the notion of ${\sigma}$-primary filters in Tarski algebras. Finally, we show that there is no non-trivial quadratic Tarski algebras on a field X with $|X|{\geq}3$.

• The Pure and Applied Mathematics
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• v.16 no.1
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• pp.59-68
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• 2009

We introduce the notion of GT-algebras as a generalization of the concept of Tarski algebras. We introduce the notion of GT-filters in GT-algebras, and we prove some properties of GT-filters.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.123-129
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• 2009

We introduce the notion of Smarandache GT-algebras, and the notion of Smarandache GT-filters of the Smarandache GT- algebra related to the Tarski algebra, and related some properties are investigated.