On closed embeddings of free topological algebras

Let $\mathcal K$ be a complete quasivariety of completely regular universal topological algebras of continuous signature $\mathcal E$ (which means that $\mathcal K$ is closed under taking subalgebras, Cartesian products, and includes all completely regular topological $\mathcal E$-algebras algebraically isomorphic to members of $\mathcal K$). For a topological space $X$ by $F(X)$ we denote the free universal $\mathcal E$-algebra over $X$ in the class $\mathcal K$. Using some extension properties of the Hartman-Mycielski construction we prove that for a closed subspace $X$ of a metrizable (more generally, stratifiable) space $Y$ the induced homomorphism $F(X)\to F(Y)$ between the respective free universal algebras is a closed topological embedding. This generalizes one result of V.Uspenskii concerning embeddings of free topological groups.Comment: 3 page

Metrizability of Clifford topological semigroups

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.Comment: 4 page

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

In this paper we search for conditions on a countably compact (pseudo-compact) topological semigroup under which: (i) each maximal subgroup $H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part $H(S)$(i.e. the union of all maximal subgroups) of the semigroup $S$ is a closed subset in $S$; (iii) the inversion $\operatorname{inv}\colon H(S)\to H(S)$ is continuous; and (iv) the projection $\pi\colon H(S)\to E(S)$, $\pi\colon x\longmapsto xx^{-1}$, onto the subset of idempotents $E(S)$ of $S$, is continuous

Algebra in superextensions of groups, I: zeros and commutativity

Given a group X we study the algebraic structure of its superextension λ(X). This is a right-topological semigroup consisting of all maximal linked systems on X endowed with the operation A∘B={C⊂X:{x∈X:x−1C∈B}∈A} that extends the group operation of X. We characterize right zeros of λ(X) as invariant maximal linked systems on X and prove that λ(X) has a right zero if and only if each element of X has odd order. On the other hand, the semigroup λ(X) contains a left zero if and only if it contains a zero if and only if X has odd order |X|≤5. The semigroup λ(X) is commutative if and only if |X|≤4. We finish the paper with a complete description of the algebraic structure of the semigroups λ(X) for all groups X of cardinality |X|≤5

Optical properties of titanium oxycarbide thin films

The optical properties of TiC x O y thin films, deposited by reactive dc magnetron sputtering at different oxygen flow, were investigated by spectroscopic ellipsometry in the energy range of 0.75–4.5 eV. The dielectric functions measured in the energy range of intraband transitions were analyzed using the classical Drude theory. These results show that free plasma energy and the damping constant of the films depend strongly on film stoichiometry and on their oxygen content. The interband contribution to the optical conductivity of these films is in good agreement with the optical conductivity obtained from first principles calculations based on density functional theory. Both the experimental and the calculated results show that it is possible to significantly modify the optical properties of titanium oxycarbide by adjusting the oxygen content.Fundação para a Ciência e a Tecnologia (FCT) – Programa Operacional “Ciência , Tecnologia, Inovação” – CONC-REEQ/443/EEI/2005, PTDC/CTM/69362 e SFRH/BD/27569/200