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Egoroff, σ, and convergence properties in some archimedean vector lattices

A. W. Hager, J. van Mill (2015)

Studia Mathematica

An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each a ₙ n ∈ ℕ c o n A ⁺ there are λ ₙ n ∈ ℕ ⊆ ( 0 , ∞ ) and a ∈ A with λₙaₙ ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent...

Embeddings of chains into chains

Vítězslav Novák, Tomáš Novotný (2005)

Discussiones Mathematicae - General Algebra and Applications

Continuity of isotone mappings and embeddings of a chain G into another chain are studied. Especially, conditions are found under which the set of points of discontinuity of such a mapping is dense in G.

Embeddings of totally ordered MV-algebras of bounded cardinality

Piotr J. Wojciechowski (2009)

Fundamenta Mathematicae

For a given cardinal number 𝔞, we construct a totally ordered MV-algebra M(𝔞) having the property that every totally ordered MV-algebra of cardinality at most 𝔞 embeds into M(𝔞). In case 𝔞 = ℵ₀, the algebra M(𝔞) is the first known MV-algebra with respect to which the deductive system for the infinitely-valued Łukasiewicz's propositional logic is strongly complete.

Engel BCI-algebras: an application of left and right commutators

Ardavan Najafi, Arsham Borumand Saeid (2021)

Mathematica Bohemica

We introduce Engel elements in a BCI-algebra by using left and right normed commutators, and some properties of these elements are studied. The notion of n -Engel BCI-algebra as a natural generalization of commutative BCI-algebras is introduced, and we discuss Engel BCI-algebra, which is defined by left and right normed commutators. In particular, we prove that any nilpotent BCI-algebra of type 2 is an Engel BCI-algebra, but solvable BCI-algebras are not Engel, generally. Also, it is proved that...

Equational spectrum of Hilbert varieties

R. Padmanabhan, Sergiu Rudeanu (2009)

Open Mathematics

We prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.

Every braid admits a short sigma-definite expression

Jean Fromentin (2011)

Journal of the European Mathematical Society

A result by Dehornoy (1992) says that every nontrivial braid admits a σ -definite expression, defined as a braid word in which the generator σ i with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a σ -definite word expression that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new...

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