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Filter theory of BL algebras. (English) Zbl 1165.03056

The paper continues the work started by M. Haveshki, A. Borumand Saeid and E. Eslami [Soft Comput. 10, No. 8, 657–664 (2006); erratum ibid. 11, No. 2, 209 (2007; Zbl 1103.03062)] on various kinds of filters in BL-algebras, namely on (positive) implicative and fantastic filters. These filters are the usual filters which are in addition closed under certain rules. Originally, Haveshki et al. proved that the quotient BL-algebras with respect to these filters are Boolean algebras, Gödel algebras, and MV-algebras, respectively.
In this paper, the authors reprove the same results in a different way. They prove that each kind of the above-mentioned filters has such properties from which the description of quotients follows easily. For instance, they show that \(F\) is an implicative filter if, and only if, \(x\to x^2\in F\) for each \(x\in F\). Analogously, \(F\) is a fantastic filter if, and only if, \(\neg\neg x\to x\in F\) for each \(x\in F\).
In summary, Haveshki et al. found alternative axiomatizations for Boolean, Gödel, and MV-algebras by means of quasi-identities. This paper reproves these facts once again.

MSC:

03G25 Other algebras related to logic
06D35 MV-algebras
06F05 Ordered semigroups and monoids

Citations:

Zbl 1103.03062
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References:

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