# Some Types of Filters in Equality Algebras

Document Type: Research Paper

Authors

1 Department of Mathematics, Shahid Beheshti University, Tehran, Iran.

2 Department of Mathematics, Payam e Noor University, Tehran, Iran.

3 Payam e Noor University

Abstract

Equality algebras were introduced by S. Jenei as a possible algebraic semantic for fuzzy type theory. In this paper, we introduce some types of filters such as (positive) implicative, fantastic, Boolean, and prime filters in equality algebras and we prove some results which determine the relation between these filters. We prove that the quotient equality algebra induced by an implicative filter is a Boolean algebra, by a fantastic filter is a commutative equality algebra, and by a prime filter is a chain, under suitable conditions. Finally, we show that positive implicative, implicative, and Boolean filters are equivalent on bounded commutative equality algebras.

Highlights

Dedicated to Bernhard Banaschewski on the occasion of his 90th birthday

Keywords

### References

[1] Borzooei, R.A., Khosravi Shoar, S., and Ameri, R., Some types of filters in MTL-algebras, Fuzzy Sets and Systems 187(1) (2012), 92-102.
[2] Cignoli, R., D'ottaviano, I., and Mundici, D., "Algebraic Foundations of Many-Valued Reasoning", Springer, Trends in Logic 7, 2000.
[3] Ciungu, L.C., Internal states on equality algebras, Soft Computing 19(4) (2015), 939-953.
[4] Esteva, F. and Godo, L., Monoidal t-normbased logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, 124(3) (2001), 271-288.
[5] H'ajek, P., "Metamathematics of Fuzzy Logic", Springer, Trends in Logic 4, 1998.
[6] Haveshki, M., Borumand Saeid, A., and Eslami, E., Some types of filters in BL algebras, Soft Computing 10(8) (2006), 657-664.
[7] Jenei, S., Equality algebras, Studia Logica 100(6) (2012), 1201-1209.
[8] Jenei, S. and K'or'odi, L., On the variety of equality algebras, Fuzzy Logic and Technology (2011), 153-155.
[9] Liu, L., On the existence of states on MTL-algebras, Inform. Sci. 220 (2013), 559-567.
[10] Liu, L. and Zhang, X., Implicative and positive implicative prefilters of EQ-algebras, J. Intell. Fuzzy Systems 26(5) (2014), 2087-2097.
[11] Nov'ak, V. and De Baets, B., EQ-algebras, Fuzzy Sets and Systems 160(20) (2009), 2956-2978.
[12] Rezaei, A., Borumand Saeid, A., and Borzooei, R.A., Some types of filters in BE-algebras, Math. Comput. Sci. 7(3) (2013), 341-352.
[13] Van Gasse, B., Deschrijver, G., Cornelis, C., and Kerre, E.E., Filters of residuated lattices and triangle algebras, Inform. Sci. 180(16) (2010), 3006-3020.
[14] Zebardast, F., Borzooei, R.A., and Aaly Kologani, M., Results on Equality algebras, Inform.  Sci. 381 (2017), 270-282.