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.


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


Zbl 1103.03062
Full Text: DOI


[1] Cignoli R, Esteva F, Godo L, Torrens A (2000) Basic fuzzy logic is the logic of continuous t-norm and their residua. Soft Comput 4:106–112 · Zbl 02181428
[2] Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht · Zbl 0937.03030
[3] Haveshki M, Saeid AB, Eslami E (2006) Some types of filters in BL algebras. Soft Comput 10:657–664 · Zbl 1103.03062
[4] Turunen E (1999) BL-algebras of basic fuzzy logic. Mathw soft comput 6:49-61 · Zbl 0962.03020
[5] Turunen E (2001) Boolean deductive systems of BL-algebras. Arch Math Logic 40:467–473 · Zbl 1030.03048
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