Algebra Universalis最新文献

A well-known result in quasigroup theory says that an associative quasigroup is a group, i.e. in quasigroups, associativity forces the existence of an identity element. The converse is, of course, far from true, as there are many, many non-associative loops. However, a remarkable theorem due to David Mumford and C.P. Ramanujam says that any projective variety having a binary morphism admitting a two-sided identity must be a group. Motivated by this result, we define a universal algebra (A; F) to be an MR-algebra if whenever a binary term function m(x, y) in the algebra admits a two-sided identity, then the reduct (A; m(x, y)) must be associative. Here we give some non-trivial varieties of quasigroups, groups, rings, fields and lattices which are MR-algebras. For example, every MR-quasigroup must be isotopic to a group, MR-groups are exactly the nilpotent groups of class 2, while commutative rings and complemented lattices are MR-algebras if and only if they are Boolean.

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander’s Subbase Theorem, which asserts that a topological space X is compact if every subbasic open cover of X admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem—whose proof depends upon Zorn’s Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander’s Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call upper Vietoris orthospaces in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday’s choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó’s choice-dependent orthospace approach to Stone duality for orthocomplemented lattices.

The system of all congruence lattices of all algebras with fixed base set A forms a lattice with respect to inclusion, denoted by (mathcal {E}_A). Let A be finite. The meet-irreducible elements of (mathcal {E}_A) are congruence lattices of monounary algebras. We assume that (A, f) has a connected subalgebra B such that B contains at least 3 cyclic elements and is meet-irreducible in ({mathcal {E}}_B) and we prove several sufficient conditions under which ({{,mathrm{Con},}}(A, f)) is meet-irreducible in ({mathcal {E}}_A).

Let A be an Archimedean f-ring with identity and assume that A is equipped with another multiplication (*) so that A is an f-ring with identity u. Obviously, if (*) coincides with the original multiplication of A then u is idempotent in A (i.e., (u^{2}=u)). Conrad proved that the converse also holds, meaning that, it suffices to have (u^{2}=u) to conclude that (*) equals the original multiplication on A. The main purpose of this paper is to extend this result as follows. Let A be a (not necessary unital) Archimedean f-ring and B be an (ell )-subgroup of the underlaying (ell )-group of A. We will prove that if B is an f-ring with identity u, then the equality (u^{2}=u) is a necessary and sufficient condition for B to be an f-subring of A. As a key step in the proof of this generalization, we will show that the set of all f-subrings of A with the same identity has a smallest element and a greatest element with respect to the inclusion ordering. Also, we shall apply our main result to obtain a well known characterization of f-ring homomorphisms in terms of idempotent elements.

Ordered by set inclusion, the retracts of a lattice L together with the empty set form a bounded poset (Ret (L)). By a grid we mean the direct product of two non-singleton finite chains. We prove that if G is a grid, then (Ret (G)) is a lattice. We determine the number of elements of (Ret (G)). Some easy properties of retracts, retractions, and their kernels called retraction congruences of (mainly distributive) lattices are found. Also, we present several examples, including a 12-element modular lattice M such that (Ret (M)) is not a lattice.

Câmpeanu and Ho (2004) determined the maximum finite state complexity of finite languages, building on work of Champarnaud and Pin (1989). They stated that it is very difficult to determine the number of maximum-complexity languages. Here we give a formula for this number. We also generalize their work from languages to functions on finite sets.

In this paper, we shed new light on the spectrum of the relation algebra we call (A_{n}), which is obtained by splitting the non-flexible diversity atom of (6_{7}) into n symmetric atoms. Precisely, show that the minimum value in (text {Spec}(A_{n})) is at most (2n^{6 + o(1)}), which is the first polynomial bound and improves upon the previous bound due to Dodd and Hirsch (J Relat Methods Comput Sci 2:18–26, 2013). We also improve the lower bound to (2n^{2} + 4n + 1), which is roughly double the trivial bound of (n^{2} + 2n + 3). In the process, we obtain stronger results regarding (text {Spec}(A_{2}) =text {Spec}(32_{65})). Namely, we show that 1024 is in the spectrum, and no number smaller than 26 is in the spectrum. Our improved lower bounds were obtained by employing a SAT solver, which suggests that such tools may be more generally useful in obtaining representation results.

A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have the natural structure of a Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.

Let ({mathcal {A}}) be a class of algebras with (I, A in {mathcal {A}}). We interpret the lattice-theoretic “strictly meet irreducible/cover” situation (B < C) in lattices of the form (S_{{mathcal {A}}}(I,A)) of all subalgebras of A containing I, where we call such (B < C) a minimum proper extension (mpe), and show that this means B is maximal in (S_{{mathcal {A}}}(I,A)) for not containing some (r in A) and C is generated by B and r. For the class ({mathcal {G}}) of groups, we determine the mpe’s in (S_{{mathcal {G}}}({0},{mathbb {Q}})) using invariants of Beaumont and Zuckerman and show that these (plus utilization of a Hamel basis) determine the mpe’s in (S_{{mathcal {G}}}({0},{mathbb {R}})). Finally, we show that the latter yield some (not all) of the minimum proper essential extensions in (mathbf {W}^{*}), the category of Archimedean (ell )-groups with strong order unit and unit-preserving (ell )-group homomorphisms.