QUOTIENT SPACE OF LMC -COMPACTIFICATION AS A SPACE OF z −FILTERS M. AKBARI TOOTKABONI Abstract. The left multiplicative continuous compactification of a semitopo- logical semigroup is the universal semigroup compactification. In this paper an internal construction of a semigroup compactification of a semitopological semigroup is constructed as a space of filters. In [6], we described an external construction of a semigroup compactification of a semitopological semigroup. 1. Introduction Stone- ˇ Cech compactifications derived from a discrete semigroup S can be con- sidered as the spectrum of the algebra B(S), the set of bounded complex-valued functions on S, or as a collection of ultrafilters on S. What is certain and indis- putable is the fact that filters play an important role in the study of Stone- ˇ Cech compactifications derived from a discrete semigroup. It seems that filters can play a role in the study of general semigroup compactifications too, (See [6] and [7]). For S a semitopological semigroup, a continuous bounded function on S is said to be in LMC (S) if its of right translates is relatively compact in CB(S) for the topol- ogy of pointwise convergence on S. In this paper we will begin with an elementary construction of semigroup compactification and we present some background about LMC(S). Also section 2 consists of an introduction to z −filters and an elementary external construction of semigroup compactification of a semitopological semigroup as a space of z −filters. In Glazer’s proof of the finite sum theorem, he introduced the notion of the sum of two ultrafilter p and q on ω, agreeing that A ∈ p + q if and only if {x : −x + A ∈ q}∈ p, (where by −x + A is meant {y : x + y ∈ A}). In section 2, we introduce the notion of the binary operation of two z −filters. Section three will be about some theorems from [2] about filters on discrete semi- group of S which are extended to a state that S be semeitopological semigroup. The scaffold of this and the following section focus on the coincidence on [2]. Therefore, all techniques and methods in [2] are established. In this section we investigate semigroup compactifications of S as spaces of z −filters. Key words and phrases. Semigroup Compactification,z-Filters, Quotient Space, m-admissible subalgebra, LMC- Compactification. 1