We find a distributive (∨, 0, 1)-semilattice Sω1 of size א1 that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: — There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with ...

Codescent morphisms are described in regular categories which satisfy the so-called strong amalgamation property. Among varieties of universal algebras possessing this property are, as is known, categories of groups, not necessarily associative rings, M -sets (for a monoid M), Lie algebras (over a field), quasi-groups, commutative quasi-groups, Steiner quasi-groups, medial quasigroups, semilatt...

We replace the group of group-like elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra w q (g). It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of w q (g) and determine the group of weak Hopf algebra automorphisms of w q (g) when q is not ...

We consider S-adic expansions associated with continued fraction algorithms, where an S-adic expansion corresponds to an infinite composition of substitutions. Recall that a substitution is a morphism of the free monoid. We focus in particular on the substitutions associated with regular continued fractions (Sturmian substitutions), and with Arnoux–Rauzy, Brun, and Jacobi–Perron (multidimension...

We study logics defined in terms of second-order monadic monoidal and groupoidal quantifiers. These are generalized quantifiers defined by monoid and groupoid word-problems, equivalently, by regular and context-free languages. We give a computational classification of the expressive power of these logics over strings with varying built-in predicates. In particular, we show that ATIME(n) can be ...

We develop the theory of transformation semigroups that have degree 2, is, act by partial functions on a finite set such inverse image points at most two elements. show graph fibers an action gives deep connection between semigroup and theory. It is known Krohn–Rhodes complexity 2 2. monoid continuous maps translational hull appropriate 0-simple semigroup. how group mapping can be considered as...

In this paper $S$ is a monoid with a left zero and $A_S$ (or $A$) is a unitary right $S$-act. It is shown that a monoid $S$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $S$-act is quasi-projective. Also it is shown that if every right $S$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that ...

we prove that if $m$ is a monoid and $a$ a finite set with more than one element‎, ‎then the residual finiteness of $m$ is equivalent to that of the monoid consisting of all cellular automata over $m$ with alphabet $a$‎.

Let R be an associative unital algebra over a field k, let p be an element of R, and let R′ = R q | pqp = p . We obtain normal forms for elements of R′, and for elements of R′-modules arising by extension of scalars from R-modules. The details depend on where in the chain pR ∩ Rp ⊆ pR ∪ Rp ⊆ pR + Rp ⊆ R the unit 1 of R first appears. This investigation is motivated by a hoped-for application to...

let r be a ring, m a right r-module and (s,≤) a strictly ordered monoid. in this paper we will show that if (s,≤) is a strictly ordered monoid satisfying the condition that 0 ≤ s for all s ∈ s, then the module [[ms,≤]] of generalized power series is a uniserial right [[rs,≤]] ]]-module if and only if m is a simple right r-module and s is a chain monoid.