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Register a new userAlgebraic representation of L-valued continuous lattices via the open filter monad
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With a complete Heyting algebra $L$ as the truth value table, we prove that the collections of open filters of stratified $L$-valued topological spaces form a monad. By means of $L$-Scott topology and the specialization $L$-order, we get that the algebras of open filter monad are precisely $L$-continuous lattices.
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Let $Sigma (X,mathbb{C})$ denote the collection of all the rings between $C^*(X,mathbb{C})$ and $C(X,mathbb{C})$. We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/$z$-ideals/$z^circ$-ideals in the rings $P(X,mathbb{C})$ in $Sigma(X,mathbb{C})$ and in their real-valued counterparts $P(X,mathbb{C})cap C(X)$. It is shown that the structure space of any such $P(X,mathbb{C})$ is $beta X$. We show that for any maximal ideal $M$ in $C(X,mathbb{C}), C(X,mathbb{C})/M$ is an algebraically closed field. We give a necessary and sufficient condition for the ideal $C_{mathcal{P}}(X,mathbb{C})$ of $C(X,mathbb{C})$ to be a prime ideal, and we examine a few special cases thereafter.
A.V.Arkhangelskii asked in 1981 if the variety $mathfrak V$ of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety $mathfrak V$ is a proper subclass of the class of all topological groups. A topological group $G$ is called $g$-sequential if for any topological group $H$ any sequentially continuous homomorphism $Gto H$ is continuous. We introduce the concept of a $g$-sequential cardinal and prove that a locally compact group is $g$-sequential if and only if its local weight is not a $g$-sequential cardinal. The product of a family of non-trivial $g$-sequential topological groups is $g$-sequential if and only if the cardinal of this family is not $g$-sequential. Suppose $G$ is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then $G$ is $g$-sequential if and only if its weight is not a $g$-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.
A proper ideal $I$ in a commutative ring with unity is called a $z^circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered field $F$ and a completely $F$-regular topological space $X$, let $C(X,F)$ be the ring of all $F$-valued continuous functions on $X$ and $B(X,F)$ the aggregate of all those functions which are bounded over $X$. An explicit formula for all the $z^circ$-ideals in $A(X,F)$ in terms of ideals of closed sets in $X$ is given. It turns out that an intermediate ring $A(X,F) eq C(X,F)$ is never regular in the sense of Von-Neumann. This property further characterizes $C(X,F)$ amongst the intermediate rings within the class of $P_F$-spaces $X$. It is also realized that $X$ is an almost $P_F$-space if and only if each maximal ideal in $C(X,F)$ is $z^circ$-ideal. Incidentally this property also characterizes $C(X,F)$ amongst the intermediate rings within the family of almost $P_F$-spaces.
In this paper, we prove that: (1) Let $f:Grightarrow H$ be a continuous $d$-open surjective homomorphism; if $G$ is an $mathbb{R}$-factorizabile paratopological group, then so is $H$. Peng and Zhangs result cite[Theorem 1.7]{PZ} is improved. (2) Let $G$ be a regular $mathbb{R}$-factorizable paratopological group; then every subgroup $H$ of $G$ is $mathbb{R}$-factorizable if and only if $H$ is $z$-embedded in $G$. This result gives out a positive answer to an question of M.~Sanchis and M.~Tkachenko cite[Problem 5.3]{ST}.
Let $M$ be an ANR space and $X$ be a homotopy dense subspace in $M$. Assume that $M$ admits a continuous binary operation $*:Mtimes Mto M$ such that for every $x,yin M$ the inclusion $x*yin X$ holds if and only if $x,yin X$. Assume also that there exist continuous unary operations $u,v:Mto M$ such that $x=u(x)*v(x)$ for all $xin M$. Given a $2^omega$-stable $mathbf Pi^0_2$-hereditary weakly $mathbf Sigma^0_2$-additive class of spaces $mathcal C$, we prove that the pair $(M,X)$ is strongly $(mathbf Pi^0_1capmathcal C,mathcal C)$-universal if and only if for any compact space $Kinmathcal C$, subspace $Cinmathcal C$ of $K$ and nonempty open set $Usubseteq M$ there exists a continuous map $f:Kto U$ such that $f^{-1}[X]=C$. This characterization is applied to detecting strongly universal Lawson semilattices.
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