t-closure Operators

Abstract

In this study, based on the knowledge of the existence of t-norms on an arbitrary given bounded lattice, we introduce t-closure operators with the help of a t-norm on the lattice and a subset of the lattice including the top element. We define two equivalence relations by using t-closure operators. The first one is on the set of all t-norms on a bounded lattice. An important class is obtained according to this relation. We define a partially order on the set of all equivalent relations given as secondly. Further, we define a set, denoted by $L^{\ast }$, and by using this set, we investigate under which conditions L can be embedded into $L_{C_{A,T}}$. We obtain a topology by using t-closure operators and examine some properties of this topology. Lastly, we generalize $T-$partial order.

Keywords:

• Closure operator
• triangular norm
• bounded lattice

2010 Mathematics Subject Clasification:

• 06B99
• 06A15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by grants from the research fund of Recep Tayyip Erdoğan University, project number: FBA-2016-603.

Notes on contributors

Mehmet Akif İnce

Mehmet Akif İnce is an assistant professor of Mathematics at Recep Tayyip Erdoğan University. He obtained his PhD from Karadeniz Technical University, Graduate School of Sciences, Department of Mathematics in 2015. He has been a member of the Department of Mathematics, Faculty of Sciences, Recep Tayyip Erdoğan University since 2011. He studies include fuzzy sets and fuzzy logic, and aggregation functions. He has published three papers in reputed international journals.

Funda Karaçal

Funda Karaçal is a professor of Mathematics at Karadeniz Technical University. She holds BSc degree in Mathematics - secondary school teacher accreditation, MSc degree in Mathematics (1996) and PhD degree in Mathematics (2001) from Karadeniz Technical University. Her research interests are fuzzy logic, triangular norms and aggregation functions. The bibliography of Professor Karaçal comprises more than 35 publications in international peer-reviewed journals, lots of presentations in international conferences.