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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.

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...

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...

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...