Fuzzy Semigroups via Semigroups

The theory of fuzzy semigroups is a branch of mathematics that arose in the early 1990s as an effort to characterise the properties of semigroups by means of the properties of their fuzzy subsystems, which include fuzzy subsemigroups and their analogues, fuzzy one (resp. two)-sided ideals, fuzzy quasi-ideals, fuzzy bi-ideals, etc. To be more precise, a fuzzy subsemigroup of a given semigroup <img src=image/13443713_01.gif> is just a ∧-prehomomorphism <img src=image/13443713_02.gif> of <img src=image/13443713_01.gif> to ([0, 1],∧). Variations of this, which correspond to the other previously mentioned fuzzy subsystems, can be obtained by imposing certain properties on f. From the work of Kuroki, Mordeson, Malik, and many of their descendants, it turns out that fuzzy subsystems play a similar role in the structure theory of semigroups that play their non-fuzzy analogues. The aim of the present paper is to show that this similarity is not coincidental. As a first step to this, we prove that there is a 1-1 correspondence between fuzzy subsemigroups of <img src=image/13443713_03.gif> and subsemigroups of a certain type of <img src=image/13443713_04.gif>. Restricted to fuzzy one-sided ideals, this correspondence identifies the above fuzzy subsystems with their analogues in <img src=image/13443713_04.gif>. Using these identifications, we prove that the characterisation of the regularity of semigroups in terms of fuzzy one-sided ideals and fuzzy quasi-ideals can be obtained as an implication of the corresponding non-fuzzy analogue. In a further step, we give new characterisations of semilattices of left simple semigroups in terms of left simple fuzzy subsemigroups, and of completely regular semigroups in terms of completely simple fuzzy subsemigroups. Both left simple fuzzy subsemigroups and completely simple fuzzy subsemigroups are defined here for the first time, and the corresponding characterisations generalise well-known characterisations of the corresponding semigroups.