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The class of 2-dimensional neat reducts is not elementary

Tarek Sayed Ahmed (2002)

Fundamenta Mathematicae

SC, CA, QA and QEA stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras and Halmos' quasipolyadic algebras with equality, respectively. Generalizing a result of Andréka and Németi on cylindric algebras, we show that for K ∈ SC,QA,CA,QEA and any β > 2 the class of 2-dimensional neat reducts of β-dimensional algebras in K is not closed under forming elementary subalgebras, hence is not elementary. Whether this result extends to higher...

The clean elements of the ring ℛ ( L )

Ali Akbar Estaji, Maryam Taha (2024)

Czechoslovak Mathematical Journal

We characterize clean elements of ℛ ( L ) and show that α ∈ ℛ ( L ) is clean if and only if there exists a clopen sublocale U in L such that 𝔠 L ( coz ( α - 1 ) ) ⊆ U ⊆ 𝔬 L ( coz ( α ) ) . Also, we prove that ℛ ( L ) is clean if and only if ℛ ( L ) has a clean prime ideal. Then, according to the results about ℛ ( L ) , we immediately get results about 𝒞 c ( L ) .

The combinatorial derivation and its inverse mapping

Igor Protasov (2013)

Open Mathematics

Let G be a group and P G be the Boolean algebra of all subsets of G. A mapping Δ: P G → P G defined by Δ(A) = {g ∈ G: gA ∩ A is infinite} is called the combinatorial derivation. The mapping Δ can be considered as an analogue of the topological derivation d: P X→ P X, A ↦ A d, where X is a topological space and A d is the set of all limit points of A. We study the behaviour of subsets of G under action of Δ and its inverse mapping ∇. For example, we show that if G is infinite and I is an ideal in...

The completely distributive lattice of machine invariant sets of infinite words

Aleksandrs Belovs, Jānis Buls (2007)

Discussiones Mathematicae - General Algebra and Applications

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.

The dimension of a variety

Ewa Graczyńska, Dietmar Schweigert (2007)

Discussiones Mathematicae - General Algebra and Applications

Derived varieties were invented by P. Cohn in [4]. Derived varieties of a given type were invented by the authors in [10]. In the paper we deal with the derived variety V σ of a given variety, by a fixed hypersubstitution σ. We introduce the notion of the dimension of a variety as the cardinality κ of the set of all proper derived varieties of V included in V. We examine dimensions of some varieties in the lattice of all varieties of a given type τ. Dimensions of varieties of lattices and all subvarieties...

The distributivity numbers of finite products of P(ω)/fin

Saharon Shelah, Otmar Spinas (1998)

Fundamenta Mathematicae

Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o. ( P ( ω ) / f i n ) n , is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).

The elementary-equivalence classes of clopen algebras of P-spaces

Brian Wynne (2008)

Fundamenta Mathematicae

Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.

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