Coz-onto Frame Maps and Some Applications

Abstract

A frame homomorphism is coz-onto if it maps the cozero part of its domain surjectively onto that of its codomain. This captures the notion of a z-embedded subspace of a topological space in a point-free setting. We give three different types of characterizations of coz-onto homomorphisms. The first is in terms of elements, the second in terms of quotients, and the last in terms of ideals. As an application of properties of coz-onto homomorphisms developed herein, we present some characterizations of F-and F -frames.

Some algebraic characterizations of F-frames

“…Then coz[Q] is an ideal of Coz L (as one checks quickly) such that whenever a ∧ b = 0 in Coz L, then a ∈ coz[Q] or b ∈ coz[Q]. Thus, by [13,Lemma 3.8], coz[Q] is a prime ideal of Coz L. Now suppose αβ ∈ Q. Then coz α ∧ coz β ∈ coz[Q].…”

“…In [13,Proposition 3.2], it is shown that if L is completely regular and h : L → M is onto, where M is Lindelöf, then h is coz-onto. Thus, by the proposition, a Lindelöf quotient of an F -frame is a C * -quotient.…”

“…Certain characterizations and other properties of F -frames are given in [13]. Lemma 4.4 in [13] states that if L is an F -frame and L → M is a coz-onto homomorphism, then M is also an F -frame. It follows from this that, for any c ∈ Coz L, we have that ↓c is an F -frame if L is an F -frame.…”

“…In the last section, we look briefly at F -frames with the view to augmenting certain results about these types of frames that were obtained in [13]. In particular, it follows from [13,Lemma 4.4] that ↓c is an F -frame for each c ∈ Coz L whenever L is an F -frame.…”

“…In particular, it follows from [13,Lemma 4.4] that ↓c is an F -frame for each c ∈ Coz L whenever L is an F -frame. Here, we strengthen this by showing that the restriction to cozero elements is not necessary (Proposition 4.1).…”