Residuated Skew Lattice with a Operation

Subject Areas : Statistics

R. Koohnavard ¹ , A. Borumand Saeid ²

1 - Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
2 - Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.

Received: 2018-02-24 Accepted : 2019-10-27 Published : 2020-01-21

Keywords: عنصر منظم, فیلتر, مشبکه کج مانده, حصار, عنصر چگال,

Abstract :

In this paper, we define hedge operation on a residuated skew lattice and investigate some its properties. We get relationships between some special sets as dense, nilpotent, idempotent, regular elements sets and their hedges. By examples, we show that hedge of a dense element is not a dense and hedge of a regular element is not a regular. Also hedge of a nilpotent element is a nilpotent and hedge of an idempotent element is not an idempotent and we show that hedge of an idempotent element is idempotent under a condition. We show that the composition of two hedges is a hedge under a condition. By examples, we show that there is no relation between the product of the hedges and the hedge of the products. H(x) is defined and some its properties are proved and is shown that hedge of set H is equal to the H itself. The relationship between H and dense, regular, nilpotent and idempotent elements is investigated too.

References:

[1] P. Jordan, "Under nichtkommutative verbande", Arch. Math, 2, 56-59, (1949).

[2] J. Leech, "Normal skew lattices", Semigroup Forum, 44 1-8, (1992).

[3] J. Leech, "Skew Boolean algebras", Algebra Univ, 27, 497-506, (1990).

[4] K. Cvetko-Vah, M. Kinyon, J. Leech, and M. Spinks, "Cancellation in skew lattices", Order, 28, 9-32, (2011).

[5] K. Cvetko-Vah, "On skew Heyting algebra", Ars Math, Contemp, 13, 37-50, (2017).

[6] J. Pita Costa, "On ideals of a skew lattice", General Algebra and Applications, 32, 5-21, (2012).

[7] J. Pita Costa, "On the coset structure of a skew lattice", Demonst Math, 44(4), 1-21, (2011).

[8] A. Borumand Saeid, R. Koohnavard, "On residuated skew lattices", Analele. Stiintific ale  Universitatii  “Ovidius”  Constanta,  27(1),  245-268, (2019).

 [9] R. Koohnavard, A. Borumand Saeid, "(Skew) filters in residuated skew lattices: Part 1, Scientific Annals of Computer Science, 28(1),  115-140, (2018).

[10] R. Koohnavard, A. Borumand Saeid, "(Skew) filters in residuated skew lattices: Part 2, Honam Mathematical Journal, 40(3), 401-431, (2018).

[11] I. Chajda, "Hedges and successors in basic algebras", Soft Comput, 15(3), 613-618, (2011).

[12] Piciu. D, "Algebras of fuzzy logic", Univ, (2007).

[13] R. Koohnavard, A. Borumand Saeid, On  Skew algebraic structures, International Conference on Recent Achievements in Mathematical Science, Yazd, Iran, (2019).