Suppose $A$ is a separable unital $𝒞\left(X\right)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_1$-injective $C^{*}$-algebra $𝒟$. We show that $A$ and $𝒞\left(X\right)\otimes 𝒟$ are isomorphic as $𝒞\left(X\right)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

Defining an (n+1)-ary superposition operation $S^n$ on the set $W_{\tau }\left(X_n\right)$ of all n-ary terms of type τ, one obtains an algebra $n-clone\tau :=(W_{\tau }\left(X_n\right);S^n,x_1,...,x_n)$ of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation $S^n$ there are different possibilities to define binary associative operations on the set $W_{\tau }\left(X_n\right)$ and on the cartesian power $W_{\tau }{\left(X_n\right)}^n$. In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative...

T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the ${ℒ}_n^m$-propositional calculus, denoted by ${\ell }_n^m$, is introduced in terms of the binary connectives $\to$ (implication), $\twoheadrightarrow$ (standard implication), $\wedge$ (conjunction), $\vee$ (disjunction) and the unary ones $f$ (negation) and $D_i$, $1\le i\le n-1$ (generalized Moisil operators). It is proved that ${\ell }_n^m$ belongs to the class of standard systems...

Let $L_c\left(X\right)=\{f\in C\left(X\right):\overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $\left|f\left(U\right)\right|\le {\aleph }_0$. In this paper, we present a pointfree topology version of $L_c\left(X\right)$, named ${ℛ}_{\ell c}\left(L\right)$. We observe that ${ℛ}_{\ell c}\left(L\right)$ enjoys most of the important properties shared by $ℛ\left(L\right)$ and ${ℛ}_c\left(L\right)$, where ${ℛ}_c\left(L\right)$ is the pointfree version of all continuous functions of $C\left(X\right)$ with countable image. The interrelation between $ℛ\left(L\right)$, ${ℛ}_{\ell c}\left(L\right)$, and ${ℛ}_c\left(L\right)$ is examined. We show that $L_c\left(X\right)\cong {ℛ}_{\ell c}\left(𝔒\left(X\right)\right)$ for any space $X$. Frames $L$ for which ${ℛ}_{\ell c}\left(L\right)=ℛ\left(L\right)$ are characterized.

Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}\left(q\right)\bowtie k\left[G\right]$ between the generalized Taft algebra $H_{m,d}\left(q\right)$ and group algebra $k\left[G\right]$ is actually the smash product $H_{m,d}\left(q\right)♯k\left[G\right]$. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$. As an application, the classification of $H_{m,d}\left(q\right)\bowtie k[C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.

Let $X$ be a zero-dimensional space and $C_c\left(X\right)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K\left(X\right)$ (resp., $C_c^{\psi }\left(X\right))$ the set of all functions in $C_c\left(X\right)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K\left(X\right)=O_c^{{\beta }_{0}X\setminus X}$ (resp., $C_c^{\psi }\left(X\right)=M_c^{{\beta }_{0}X\setminus {\upsilon }_{0}X}$), where ${\beta }_{0}X$ is the Banaschewski compactification of $X$ and ${\upsilon }_{0}X$ is the $\mathbb{N}$-compactification of $X$. This implies that for an $\mathbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c\left(X\right)$ is equal to $C_c^K\left(X\right)$, i.e., $M_c^{{\beta }_{0}X\setminus X}=C_c^K\left(X\right)$. By applying...

For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$; the $c$-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$. Its maximal cones are naturally indexed by the $c$-sortable elements of $W$. The main result of this paper is that the known bijection cl${}_c$ between $c$-sortable elements and $c$-clusters induces a combinatorial isomorphism of fans. In particular, the $c$-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...

Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{sg},I^{sg})$ and $(Q^g,I^g)$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K{Q}^{sg}/⟨I^{sg}⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K{Q}^g/⟨I^g⟩$. As a corollary, we find that $gldimK{Q}^{sg}/⟨I^{sg}⟩<\infty$ if and only if $gldimKQ/⟨I⟩<\infty$ if and only if $gldimK{Q}^g/⟨I^g⟩<\infty$.

Let $𝒜={\left\{A_t\right\}}_{t\in G}$ and $ℬ={\left\{B_t\right\}}_{t\in G}$ be $C^{*}$-algebraic bundles over a finite group $G$. Let $C={⨁}_{t\in G}{A}_t$ and $D={⨁}_{t\in G}{B}_t$. Also, let $A=A_e$ and $B=B_e$, where $e$ is the unit element in $G$. We suppose that $C$ and $D$ are unital and $A$ and $B$ have the unit elements in $C$ and $D$, respectively. In this paper, we show that if there is an equivalence $𝒜-ℬ$-bundle over $G$ with some properties, then the unital inclusions of unital $C^{*}$-algebras $A\subset C$ and $B\subset D$ induced by $𝒜$ and $ℬ$ are strongly Morita equivalent. Also, we suppose that $𝒜$ and $ℬ$ are saturated and that $A^{\text{'}}\cap C=𝐂1$. We show that...

We investigate derived equivalences between subalgebras of some $\Phi$-Auslander-Yoneda algebras from a class of $n$-angles in weakly $n$-angulated categories. The derived equivalences are obtained by transferring subalgebras induced by $n$-angles to endomorphism algebras induced by approximation sequences. Then we extend our constructions in T. Brüstle, S. Y. Pan (2016) to $n$-angle cases. Finally, we give an explicit example to illustrate our result.

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space $𝐗=(X,T)$ is countably compact if and only if it is countably subcompact relative to $T$. (iii) For every metrizable space $𝐗=(X,T)$, the following are equivalent: (a) $𝐗$ is compact; (b) for every open filter $ℱ$ of $𝐗$, $\bigcap \{\overline{F}:F\in ℱ\}\ne \varnothing$; (c) $𝐗$ is subcompact relative to $T$. We also show: (iv) The negation of each of the statements, (a) every countably subcompact...

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme ${𝔅}_n$ of finite type over $\mathbb{Z}$, with geometrically connected fibers. It is smooth if and only if $n\le 3$. It is reducible if $n\ge 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$). The relative dimension of ${𝔅}_n$ is $\frac{2}{27}{n}^3+O\left(n^{8/3}\right)$. The subscheme parameterizing étale algebras is isomorphic to ${GL}_n/S_n$, which is of dimension only $n^2$. For $n\ge 8$, there exist algebras that are not limits of étale algebras. The dimension...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...