A theorem of Gleason states that every compact space admits a projective cover. More generally, in the category of topological spaces with continuous maps, covers exist with respect to the full subcategory of extremally disconnected spaces. Such a cover of a space is called its absolute. We prove that the absolute exists within the category of schematic spaces, i.e. the spaces underlying a scheme. For a schematic space, we use the absolute to generalize Bourbaki's concept of irreducible component,...

Effect algebras are very natural logical structures as carriers of probabilities and states. They were introduced for modeling of sets of propositions, properties, questions, or events with fuzziness, uncertainty or unsharpness. Nevertheless, there are effect algebras without any state, and questions about the existence (for non-modular) are still unanswered. We show that every Archimedean atomic lattice effect algebra with at most five blocks (maximal MV-subalgebras) has at least one state, which...

Given a basis of pseudoidentities for a pseudovariety of ordered semigroups containing the 5-element aperiodic Brandt semigroup $B_2$, under the natural order, it is shown that the same basis, over the most general graph over which it can be read, defines the global. This is used to show that the global of the pseudovariety of level $3/2$ of Straubing-Thérien’s concatenation hierarchy has infinite vertex rank.

Given a basis of pseudoidentities for a pseudovariety of ordered semigroups containing the 5-element aperiodic Brandt semigroup B2, under the natural order, it is shown that the same basis, over the most general graph over which it can be read, defines the global. This is used to show that the global of the pseudovariety of level 3/2 of Straubing-Thérien's concatenation hierarchy has infinite vertex rank.

We attach to each $\langle 0,\vee \rangle$-semilattice $S$ a graph $G_S$ whose vertices are join-irreducible elements of $S$ and whose edges correspond to the reflexive dependency relation. We study properties of the graph $G_S$ both when $S$ is a join-semilattice and when it is a lattice. We call a $\langle 0,\vee \rangle$-semilattice $S$ particle provided that the set of its join-irreducible elements satisfies DCC and join-generates $S$. We prove that the congruence lattice of a particle lattice is anti-isomorphic to the lattice of all hereditary subsets of...

$W$ denotes the class of abstract algebras of the title (with homomorphisms preserving unit). The familiar $H,S,$ and $P$ from universal algebra are here meant in $W$. $\mathbb{Z}$ and $ℝ$ denote the integers and the reals, with unit 1, qua$W$-objects. $V$ denotes a non-void finite set of positive integers. Let $𝒢\subseteq W$ be non-void and not $\left\{\right\{0\left\}\right\}$. We show(1)$HSP𝒢=HSP(HS𝒢\cap Sℝ)$, and(2)$W\ne 𝒢=HSP𝒢$ if and only if $\exists V(𝒢=HSP\left\{\frac{1}{v}\mathbb{Z}\right|v\in V\}).$Our proofs are, for the most part, simple calculations. There is no real use of methods of universal algebra (e.g., free objects), and only one restricted...

We prove the $*$–version of the Joly–Becker theorem: a skew field admits a $*$–ordering of level $n$ iff it admits a $*$–ordering of level $n\ell$ for some (resp. all) odd $\ell \in \mathbb{N}$. For skew fields with an imaginary unit and fields stronger results are given: a skew field with imaginary unit that admits a $*$–ordering of higher level also admits a $*$–ordering of level $1$. Every field that admits a $*$–ordering of higher level admits a $*$–ordering of level $1$ or $2$

The main result of the paper characterizes continuous local derivations on a class of commutative Banach algebra $A$ that all of its squares are positive and satisfying the following property: Every continuous bilinear map $\Phi$ from $A\times A$ into an arbitrary Banach space $B$ such that $\Phi (a,b)=0$ whenever $ab=0$, satisfies the condition $\Phi (ab,c)=\Phi (a,bc)$ for all $a,b,c\in A$.