Abstract. Let X be a nonempty set, and let F = {Y i : i ∈ I} be a family of nonempty subsets of X with the properties that X = i∈I Y i , and Y i ∩ Y j = ∅ for all i, j ∈ I with i = j. Let ∅ = J ⊆ I, and let T (J )Next, we define two congruence relations χ and χ on T (J )We begin this paper by studying the regularity of the quotient semigroups T (J )

In this paper, we make use of the notion of the character of a transformation on a fixed set X , provided by Purisang and Rakbud in 2016, and the notion of a Δ -structure on X , provided by Magill Jr. and Subbiah in 1974, to define a sub-semigroup of the full-transformation semigroup T X . We also define a sub-semigroup of that semigroup. The regularity of those two semigroups is also studied.

Let [Formula: see text] be fixed. For each [Formula: see text], we define the set [Formula: see text] of sequences of infinite complex matrices and the norm [Formula: see text] on [Formula: see text] analogously to the classical sequence space [Formula: see text] and the [Formula: see text]-norm as follows: [Formula: see text] and [Formula: see text] where [Formula: see text] is referred to as the operator norm on [Formula: see text]. We first study the completeness of sequence space [Formula: see text] equipped with the norm [Formula: see text]. Some duality results are also discussed. The aim of this paper is to show that for the case where [Formula: see text], the dual [Formula: see text] of [Formula: see text] can be decomposed as an [Formula: see text] direct-sum of its two closed subspaces. This is done by a way analogous to the theorem of Dixmier on decomposing the dual [Formula: see text] of [Formula: see text].

Abstract. Let K be a compact Hausdorff space, A a commutative complex Banach algebra with identity and C (A ) the set of characters of A . In this note, we study the class of functions f : K → A such that Ω A • f is continuous, where Ω A denotes the Gelfand representation of A . The inclusion relations between this class, the class of continuous functions, the class of bounded functions and the class of weakly continuous functions relative to the weak topology σ(A , C (A )), are discussed. We also provide some results on its completeness under the norm defined by |f | = Ω A • f ∞ .

In this paper, we introduce the notion of a solvable triple of binary relations on a set. This notion generalizes the notion of a regular relation and all other notions that are variants of the notion of the regularity, defined previously by many people. We also give some characterizations of the solvability of a triple of relations and use this to study Green’s relations on the monoid of binary relations on a set.