Algebra Universalis最新文献

In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor?

Using the theory on infinite primes of fields developed by Harrison in [2], the necessary and sufficient conditions are proved for real number fields to be (O^{*})-fields, and many examples of (O^{*})-fields are provided.

We construct non-abelian totally ordered groups of matrices of finite Archimedean rank using the group of o-automorphisms of direct sums of copies of the reals ordered anti-lexicographically. We also prove that each of these o-groups is divisible, and provide, for every (n>2), a specific formula to find the n-th root of every element of such group. Finally, we construct an example of a non-commutative totally ordered ring.

The Mal’tsev product of two varieties of the same similarity type is not in general a variety, because it can fail to be closed under homomorphic images. In the previous paper we provided new sufficient conditions for such a product to be a variety. In this paper we extend that result by weakening the assumptions regarding the two varieties. We also explore the various special cases of our new result and provide a number of examples of its application.

Suppose p(x, y, z) and q(x, y, z) are terms. If there is a common “ancestor” term (s(z_{1},z_{2},z_{3},z_{4})) specializing to p and q through identifying some variables

then the equation

is a trivial consequence. In this note we show that for lattice terms, and more generally for terms of lattice-ordered algebras, a converse is true, too. Given terms p, q,  and an equation

where ({u_{1},ldots ,u_{m}}={v_{1},ldots ,v_{n}},) there is always an “ancestor term” (s(z_{1},ldots ,z_{r})) such that (p(x_{1},ldots ,x_{m})) and (q(y_{1},ldots ,y_{n})) arise as substitution instances of s,  whose unification results in the original equation ((*)). In category theoretic terms the above proposition, when restricted to lattices, has a much more concise formulation:Free-lattice functors weakly preserve pullbacks of epis. Finally, we show that weak preservation is all that we can hope for. We prove that for an arbitrary idempotent variety ({{mathcal {V}}}) the free-algebra functor (F_{{mathcal {V}}}) will not preserve pullbacks of epis unless ({{mathcal {V}}}) is trivial (satisfying (xapprox y)) or ({{mathcal {V}}}) contains the “variety of sets” (where all operations are implemented as projections).

We study the varieties with (vec {0}) and (vec {1}) where factor congruences are definable by existential formulas parameterized by central elements. This continues previous work on equational definability of factor congruences.

A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice (mathrm{N }A) that is isomorphic to the system ({mathcal {N}}A) of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed.

We extend the validity of Kiss’s characterization of “([alpha ,beta ]=0)” from congruence modular varieties to varieties with a difference term. This fixes a recently discovered gap in our paper Kearnes et al. (Trans Am Math Soc 368:2115–2143, 2016). We also prove some related properties of Kiss terms in varieties with a difference term.

We prove that for any act X over a finite semigroup S, the congruence lattice ({{,mathrm{Con},}}X) embeds the lattice ({{,mathrm{Eq},}}M) of all equivalences of an infinite set M if and only if X is infinite. Equivalently: for an act X over a finite semigroup S, the lattice ({{,mathrm{Con},}}X) satisfies a non-trivial identity if and only if X is finite. Similar statements are proved for an act with zero over a completely 0-simple semigroup ({mathcal {M}}^0(G,I,Lambda ,P)) where (|G|,|I| <infty ). We construct examples that show that the assumption (|G|,|I| <infty ) is essential.