Bulletin of the Iranian Mathematical Society , Jul 1, 2024
We introduce zero-dimensionally embedded (ZDE) sublocales as those sublocales
S with the propert... more We introduce zero-dimensionally embedded (ZDE) sublocales as those sublocales
S with the property that the ambient frame has a basis, elements of which induce
open sublocales whose frontiers miss S. This notion is stronger than the traditional
zero-dimensionality of a sublocale. A compactification of a frame is perfect if its
associated right adjoint preserves disjoint binary joins. Herein, the class of rim-perfect
compactifications of frames is introduced, and we show that it contains all the perfect
ones. Indeed, not every rim-perfect compactification is perfect, but compactifications
with a ZDE remainder do not distinguish between rim-perfectness and perfectness.
The Freudenthal compactification has a ZDE remainder. We show that a frame L is
rim-compact if and only if L has a compactification with a ZDE remainder. Several
results concerning perfect compactifications and ZDE remainders are provided.
In this note we present an investigation of J -spaces and other types of spaces related to J -spa... more In this note we present an investigation of J -spaces and other types of spaces related to J -spaces. We characterise these spaces using the notion of relatively connected subsets. For a completely regular J -space X, we give a description of the βX-extension of idempotent elements of C(X) and characterise completely regular J -spaces using idempotent elements of some subrings of C(X). A brief study of relatively connected subsets which leads to the definition of C-normal spaces is given. We show that this new class of C-normal spaces contains the class of normal spaces as a proper subclass. Michael proved that βX \ X is relatively connected in βX if and only if X is a completely regular J -space. We further establish conditions which characterise completely regular J -spaces via some closed subsets of βX.
We examine N-star compactifications of non-compact regular continuous frames and provide conditio... more We examine N-star compactifications of non-compact regular continuous frames and provide conditions under which these kind of compactifications are perfect. Some properties and characterisations of such compactifications are established for the cases where N ∈ { 1 , 2 } . Connectedness of the remainder of a non-compact regular continuous frame in these compactifications is also investigated.
We introduce the definition of h - perfect elements relative to a compactification $$h:M\longrigh... more We introduce the definition of h - perfect elements relative to a compactification $$h:M\longrightarrow L$$ h : M ⟶ L and show that if a collection of all such elements is a basis, then the remainder of a frame in this compactification is zero-dimensional. This concept yields what we call a full $$\pi $$ π -compact basis for rim-compact frames. Compactifications arising from full $$\pi $$ π -compact bases are investigated. We show that the Freudenthal compactification is the smallest perfect compactification and that its basis is full. Also, we exhibit the one-to-one correspondence between the set of all full $$\pi $$ π -compact bases and the set of all $$\pi $$ π -compactifications of a rim-compact frame L .
We show that any compactification of a regular frame can be realised as a frame which is freely g... more We show that any compactification of a regular frame can be realised as a frame which is freely generated on a suitable meet semilattice subject to certain relations, and we provide familiar examples of compactifications which can be regarded as such frames.
Bulletin of the Iranian Mathematical Society , Jul 1, 2024
We introduce zero-dimensionally embedded (ZDE) sublocales as those sublocales
S with the propert... more We introduce zero-dimensionally embedded (ZDE) sublocales as those sublocales
S with the property that the ambient frame has a basis, elements of which induce
open sublocales whose frontiers miss S. This notion is stronger than the traditional
zero-dimensionality of a sublocale. A compactification of a frame is perfect if its
associated right adjoint preserves disjoint binary joins. Herein, the class of rim-perfect
compactifications of frames is introduced, and we show that it contains all the perfect
ones. Indeed, not every rim-perfect compactification is perfect, but compactifications
with a ZDE remainder do not distinguish between rim-perfectness and perfectness.
The Freudenthal compactification has a ZDE remainder. We show that a frame L is
rim-compact if and only if L has a compactification with a ZDE remainder. Several
results concerning perfect compactifications and ZDE remainders are provided.
In this note we present an investigation of J -spaces and other types of spaces related to J -spa... more In this note we present an investigation of J -spaces and other types of spaces related to J -spaces. We characterise these spaces using the notion of relatively connected subsets. For a completely regular J -space X, we give a description of the βX-extension of idempotent elements of C(X) and characterise completely regular J -spaces using idempotent elements of some subrings of C(X). A brief study of relatively connected subsets which leads to the definition of C-normal spaces is given. We show that this new class of C-normal spaces contains the class of normal spaces as a proper subclass. Michael proved that βX \ X is relatively connected in βX if and only if X is a completely regular J -space. We further establish conditions which characterise completely regular J -spaces via some closed subsets of βX.
We examine N-star compactifications of non-compact regular continuous frames and provide conditio... more We examine N-star compactifications of non-compact regular continuous frames and provide conditions under which these kind of compactifications are perfect. Some properties and characterisations of such compactifications are established for the cases where N ∈ { 1 , 2 } . Connectedness of the remainder of a non-compact regular continuous frame in these compactifications is also investigated.
We introduce the definition of h - perfect elements relative to a compactification $$h:M\longrigh... more We introduce the definition of h - perfect elements relative to a compactification $$h:M\longrightarrow L$$ h : M ⟶ L and show that if a collection of all such elements is a basis, then the remainder of a frame in this compactification is zero-dimensional. This concept yields what we call a full $$\pi $$ π -compact basis for rim-compact frames. Compactifications arising from full $$\pi $$ π -compact bases are investigated. We show that the Freudenthal compactification is the smallest perfect compactification and that its basis is full. Also, we exhibit the one-to-one correspondence between the set of all full $$\pi $$ π -compact bases and the set of all $$\pi $$ π -compactifications of a rim-compact frame L .
We show that any compactification of a regular frame can be realised as a frame which is freely g... more We show that any compactification of a regular frame can be realised as a frame which is freely generated on a suitable meet semilattice subject to certain relations, and we provide familiar examples of compactifications which can be regarded as such frames.
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Papers by Simo Mthethwa
S with the property that the ambient frame has a basis, elements of which induce
open sublocales whose frontiers miss S. This notion is stronger than the traditional
zero-dimensionality of a sublocale. A compactification of a frame is perfect if its
associated right adjoint preserves disjoint binary joins. Herein, the class of rim-perfect
compactifications of frames is introduced, and we show that it contains all the perfect
ones. Indeed, not every rim-perfect compactification is perfect, but compactifications
with a ZDE remainder do not distinguish between rim-perfectness and perfectness.
The Freudenthal compactification has a ZDE remainder. We show that a frame L is
rim-compact if and only if L has a compactification with a ZDE remainder. Several
results concerning perfect compactifications and ZDE remainders are provided.
S with the property that the ambient frame has a basis, elements of which induce
open sublocales whose frontiers miss S. This notion is stronger than the traditional
zero-dimensionality of a sublocale. A compactification of a frame is perfect if its
associated right adjoint preserves disjoint binary joins. Herein, the class of rim-perfect
compactifications of frames is introduced, and we show that it contains all the perfect
ones. Indeed, not every rim-perfect compactification is perfect, but compactifications
with a ZDE remainder do not distinguish between rim-perfectness and perfectness.
The Freudenthal compactification has a ZDE remainder. We show that a frame L is
rim-compact if and only if L has a compactification with a ZDE remainder. Several
results concerning perfect compactifications and ZDE remainders are provided.