# Strong Fuzzy Topological Groups

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 38, (2008), **Pages** 187-195

**V.L.G. Nayagam**

Department of Mathematics

National Institute of Technology

Tiruchirapalli

INDIA

mailto:velulakshmanan@nitt.edu

**D. Gauld**

Department of Mathematics

University of Auckland

Auckland

NEW ZEALAND

**G. Venkateshwari**

Department of Mathematics

Sacs MAVMM Engineering College

Madurai

INDIA

**G. Sivaraman**

Department of Mathematics

Anna University

Chennai

INDIA

**Abstract** Following the introduction of fuzzy sets in 1965, a notion of fuzzy topological group was
proposed by Foster in 1979: essentially he took a group and furnished it with a fuzzy
topological structure. An equivalent notion of fuzzy topological group was introduced by
Ma and Yu in 1984 by replacing points by fuzzy points. Recently two of the coauthors have introduced the notion of the topology
induced on the set of all fuzzy singletons by the fuzzy topology. In this paper we extend the
notion of fuzzy topological group by allowing the points of our strong fuzzy topological
groups to be fuzzy singletons of a given group and using the induced topology. We study
properties of strong fuzzy topological groups, analysing such entities as its connection
with the previous notions, subgroups, images and products of strong fuzzy topological
groups.

**Keywords** fuzzy topological spaces, fuzzy subgroups, fuzzy left coset,
fuzzy Hausdorff space, translation invariant topology, fuzzy topological group.

**Classification** (MSC2000) 54H11, 54A40, 20N25.