Continuous Functions in Hashimoto Topologies and Their Algebraic Properties

In the paper we consider the Hashimoto topologies on the interval $$[0,1]$$ [ 0 , 1 ] as well as on $$\mathbb {R}$$ R , which are connected with the natural topology on $$\mathbb {R}$$ R and with some important and well known $$\sigma $$ σ -ideals in $$\mathcal {P}(\mathbb {R})$$ P ( R ) . We study the families of continuous functions $$f:[0,1]\rightarrow \mathbb {R}$$ f : [ 0 , 1 ] → R with respect to the same Hashimoto topology $$\mathcal {H}(\mathcal {I})$$ H ( I ) (connected with the $$\sigma $$ σ -ideal $$\mathcal {I}$$ I ) on the domain and on the range of the considered functions. We show that inside common parts and differences of some such families we can find large ($$\mathfrak {c}$$ c -generated) free algebras. Some of constructed algebras appear dense in the algebra of the functions which are continuous in the usual sense.

Equivalential algebras with conjunction on the regular elements

We introduce the definition of the three-element equivalential algebra R with conjunction on the regular elements. We study the variety generated by R and prove the Representation Theorem. Then, we construct the finitely generated free algebras and compute the free spectra in this variety.

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].

A note on 3×3-valued Łukasiewicz algebras with negation

In 2004, C. Sanza, with the purpose of legitimizing the study of $$n\times m$$-valued Łukasiewicz algebras with negation (or $$NS_{n\times m}$$-algebras) introduced $$3\times 3$$-valued Łukasiewicz algebras with negation. Despite the various results obtained about $$NS_{n\times m}$$-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated $$NS_{3 \times 3}$$-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice $$\Lambda$$(NS$$_{3\times 3}$$) of all subvarieties of NS$$_{3\times 3}$$ and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of NS_$${3\times 3}$$.

Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.

Minimal varieties of associative algebras and transcendental series

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number [Formula: see text] of minimal varieties of given exponent [Formula: see text] is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit [Formula: see text] exists and can be expressed as the positive solution of an equation [Formula: see text] where [Formula: see text] is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank [Formula: see text]. It follows from classical results on lacunary power series that the generating function of the sequence [Formula: see text], [Formula: see text], is transcendental. With the same approach we construct examples of free graded semigroups [Formula: see text] with the following property. If [Formula: see text] is the number of elements of degree [Formula: see text] of [Formula: see text], then the limit [Formula: see text] exists and is transcendental.