Ultra deductive systems and (nilpotent) Boolean elements in hoops

Document Type : Original Article


1 Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

2 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea

3 Hatef Higher Education Institute, Zahedan, Iran


In this paper, first we define the concept of nilpotent element on a hoop H, study some properties of them and investigate the relation with ultra deductive systems. Then by using this notion, we introduce cyclic hoops and prove that every cyclic hoop has a unique generator and is a local MV-algebra. In the follows, we introduce the notion of Boolean elements on hoops and investigate some of their properties and relation among Boolean elements with ultra deductive systems and nilpotent elements. Finally, we introduce a functor between the category of hoops and category of Boolean elements of them.


[1] M. Aaly Kologani, R.A. Borzooei, On ideal theory on hoops, Mathematica Bohemica, 145(2) (2019),
[2] R.A. Borzooei, M. Aaly Kologani, Filter theory of hoop algebras, Journal of Advanced Research in Pure
Mathematics, 6(4) (2014), 72–86.
[3] R.A. Borzooei, M. Aaly Kologani, Stabilizer topology of hoops, Algebraic Structures and Their Applications,
1(1) (2014), 35–48.
[4] R.A. Borzooei, M. Aaly Kologani, Local and perfect semihoops, Journal of Intelligent and Fuzzy Systems,
29 (2015), 223–234.
[5] R.A. Borzooei, M. Aaly Kologani, Results on hoops, Journal of Algebraic Hyperstructures and Logical
Algebras, 1(1) (2020), 61–77.
[6] R.A. Borzooei, O. Zahiri, Radical and it’s applications in BCH-algebras, Iranian Journal of Mathematical
Sciences and Informatics, 8(1) (2013), 15–29.
[7] B. Bosbach, Komplementäre Halbgruppen. Axiomatik und Arithmetik, Fundamenta Mathematicae,
64(3) (1969), 257–287.
[8] B. Bosbach, Komplementäre Halbgruppen. Kongruenzen und Quatienten, Fundamenta Mathematicae,
69(1) (1970), 1–14.
[9] F. Esteva, L. Godo, Monoidal t-norm based logic, towards a logic for left-continuous t-norms, Fuzzy
Sets and Systems, 124(3) (2001), 271–288.
[10] G. Georgescu, L. Leustean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued Logic and Soft
Computing, 11(1-2) (2005), 153–184.
[11] G. Georgescu, C. Muresan, Boolean lifting property for residuated lattices, Soft Computing, 18 (2014),
[12] P. Hájek, Mathematics of fuzzy logic, Kluwer Academic Publishers, Dordrecht, 1998.
[13] M. Kondo, W.A. Dudek, Filter theory of BL-algebras, Soft Computing, 12(5) (2008), 419–423.
[14] T. Kowalski, H. Ono, Residuated lattices: An algebraic glimpse at logic without contraction, Japan
Advanced Institute of Science and Technology, 2001.
[15] E. Mohammadzadeh, R.A. Borzooei, Engel, nilpotent and solvable BCI-algebras, An. St. Univ. Ovidius
Constantsa, 27(1) (2019), 169–192.
[16] C. Muresan, Dense elements and classes of residuated lattices, arXiv:0901.1630vl, (2009).
[17] F. Xie, H. Liu, Ideals in pseudo-hoop algebras, Journal of Algebraic Hyperstructures and Logical Algebras,
1(4) (2020), 39–53.