We completely determine all distributive, codistributive, standard, costandard, and neutral elements in the lattice of overcommutative semigroup varieties, thus correcting a gap contained in [5].

Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $mathscr{B}$ (resp. $varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with  $mathscr{B}$ (resp. $varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbase...

We investigate the lattice of machine invariant classes. This is an infinite completely distributive lattice but it is not a Boolean lattice. The length and width of it is c. We show the subword complexity and the growth function create machine invariant classes.

and Applied Analysis 3 such that a ∈⇓ d. Thus d ∈⇑ a ⊆ U. Therefore D ∩ U ̸ = 0, hence U ∈ σ(P). This proves τ(P) ⊆ σ(P). (2) Now assume that P is strongly continuous. Then P is continuous. In a strongly continuous lattice, the relation ≪ and ⇐ are the same by the Theorem 2.5 of [5]. Also by [2] in a continuous lattice, every Scott open set A satisfies the condition A = ↑≪A, it follows that ever...

in this paper, countable compactness and the lindel¨of propertyare defined for l-fuzzy sets, where l is a complete de morgan algebra. theydon’t rely on the structure of the basis lattice l and no distributivity is requiredin l. a fuzzy compact l-set is countably compact and has the lindel¨ofproperty. an l-set having the lindel¨of property is countably compact if andonly if it is fuzzy compact. ...