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In this paper we study the reflexive subobject lattices and reflexive endomorphism algebras in a concrete category. For the category of sets and mappings, a complete characterization for both reflexive subobject lattices and reflexive endomorphism algebras is obtained. Some partial results are also proved for the category of abelian groups.

A dcpo $P$ is continuous if and only if the lattice $C\left(P\right)$ of all Scott-closed subsets of $P$ is completely distributive. However, in the case where $P$ is a non-continuous dcpo, little is known about the order structure of $C\left(P\right)$. In this paper, we study the order-theoretic properties of $C\left(P\right)$ for general dcpo’s $P$. The main results are: (i) every $C\left(P\right)$ is C-continuous; (ii) a complete lattice $L$ is isomorphic to $C\left(P\right)$ for a complete semilattice $P$ if and only if $L$ is weak-stably C-algebraic; (iii) for any two complete semilattices...

For a topological space $X$, let $S\left(X\right)$ denote the set of all closed subsets in $X$, and let $C\left(X\right)$ denote the set of all continuous maps $f:X\to X$. A family $𝒜\subseteq S\left(X\right)$ is called reflexive if there exists $𝒞\subseteq C\left(X\right)$ such that $𝒜=\{A\in S(X):f(A)\subseteq A$ for every $f\in 𝒞\}$. Every reflexive family of closed sets in space $X$ forms a sub complete lattice of the lattice of all closed sets in $X$. In this paper, we continue to study the reflexive families of closed sets in various types of topological spaces. More necessary and sufficient conditions for certain families of closed...

We lift the notion of quasicontinuous posets to the topology context, called quasicontinuous spaces, and further study such spaces. The main results are: (1) A $T_0$ space $(X,\tau )$ is a quasicontinuous space if and only if $SI\left(X\right)$ is locally hypercompact if and only if $({\tau }_{SI},\subseteq )$ is a hypercontinuous lattice; (2) a $T_0$ space $X$ is an $SI$-continuous space if and only if $X$ is a meet continuous and quasicontinuous space; (3) if a $C$-space $X$ is a well-filtered poset under its specialization order, then $X$ is a quasicontinuous space...