Abstract We examine the problem of projecting subsets a commutative, positively ordered monoid into an o -ideal. prove that to this end one may restrict sufficient subset, for whose cardinality we provide explicit upper bound. Several applications set functions, vector lattices and other more structures are provided.

The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let M be a reductive monoid with unit group G, Borel subgroup B and Weyl group W . We study the partially ordered set of B×Borbits (with respect to Zariski closure inclusion) within a G × G-orbit of M . This is the same as studying a W ×W -or...

1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ?) where G is a nonempty set and ? is a binary operation on G satisfying: (G1) a ? (b ? c) = (a ? b) ? c, ∀a, b, c ∈ G. A semigroup G is a monoid if it also satisfies: (G2) G has an element e (sometimes denoted by 1G, called the identity of G), such that e ? a = a ? e = a, ∀a ∈ G. A monoid is a group if (G3) below is satisfied...

In 1970, R. S. Cohen and Janusz A. Brzozowski introduced a hierarchy of star-free languages called the dot-depth hierarchy. This hierarchy and its generalisations, together with the problems attached to them, had a long-lasting influence on the development of automata theory. This survey article reports on the numerous results and conjectures attached to this hierarchy. This paper is a follow-u...

A lattice ordered monoid is a structure 〈L;⊕, 0L;≤〉 where 〈L;⊕, 0L〉 is a monoid, 〈L;≤〉 is a lattice and the binary operation ⊕ distributes over finite meets. If M ∈ R-Mod then the set ILM of all hereditary pretorsion classes of σ[M ] is a lattice ordered monoid with binary operation given by α :M β := {N ∈ σ[M ] | there exists A ≤ N such that A ∈ α and N/A ∈ β}, whenever α, β ∈ ILM (the subscri...

The concept of residuated relational systems ordered under a quasi-order relation was introduced in 2018 by S. Bonzio and I. Chajda as structure $\mathfrak{A} = \langle A, \cdot, \rightarrow, 1, \preccurlyeq \rangle$, where $(A,\cdot)$ is commutative semigroup with the identity $1$ top element this monoid $\preccurlyeq$. In 2020, author analyzed concepts filters type algebraic structures. addit...

A class K of similar algebras is said to have the finite embeddability property (briefly, the FEP) if every finite subset of an algebra in K can be extended to a finite algebra in K, with preservation of all partial operations. If a finitely axiomatized variety or quasivariety of finite type has the FEP, then its universal first order theory is decidable, hence its equational and quasi-equation...

The class of Guaranteed Scoring Games (GS) is the class of all two-player scoring combinatorial games with the property that Normal-play games (Conway et. al.) are order embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Fi...

The concepts of fuzzy source and fuzzy successor operators for an L-fuzzy automaton (L is a latticeordered monoid) are introduced, which turn out to be L-fuzzy closure operators. When L is a quantale, these operators introduce L-fuzzy topologies. These observations are then used to give topological characterization of the separatedness and connectedness properties of an L-fuzzy automaton. c ©20...

Intrinsic complete characterizations of constructive, context-free and regular languages have been formulated by means of configurations of languages. The definition of a semiconfiguration is given here by generalizing the definition of a configuration. By means of semiconfigurations, an intrinsic complete characterization of context-sensitive languages is formulated. 1. Languages and generaliz...