Algebra Universalis最新文献

In this paper, we study projective algebras in varieties of (bounded) commutative integral residuated lattices. We make use of a well-established construction in residuated lattices, the ordinal sum, and the order property of divisibility. Via the connection between projective and splitting algebras, we show that the only finite projective algebra in (mathsf {{FL}_{ew}}) is the two-element Boolean algebra. Moreover, we show that several interesting varieties have the property that every finitely presented algebra is projective, such as locally finite varieties of hoops. Furthermore, we show characterization results for finite projective Heyting algebras, and finitely generated projective algebras in locally finite varieties of bounded hoops and BL-algebras. Finally, we connect our results with the algebraic theory of unification.

The algebraic geometry of a universal algebra ({textbf{A}}) is defined as the collection of solution sets of systems of term equations. Two algebras ({textbf{A}}_1) and ({textbf{A}}_2) are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set A with (|A|) there are countably many algebraically inequivalent Mal’cev algebras and that on a finite set A with (|A|) there are continuously many algebraically inequivalent algebras.

Motivated by the classical work of Halmos on functional monadic Boolean algebras, we derive three basic sup-semilattice constructions, among other things, the so-called powersets and powerset operators. Such constructions are extremely useful and can be found in almost all branches of modern mathematics, including algebra, logic, and topology. Our three constructions give rise to four covariant and two contravariant functors and constitute three adjoint situations we illustrate in simple examples.

We explore a pointfree theory of bitopological spaces (that is, sets equipped with two topologies). In particular, here we regard finitary biframes as duals of bitopological spaces. In particular for a finitary biframe (mathcal {L}) the ordered collection of all its pointfree bisubspaces (i.e. its biquotients) is studied. It is shown that this collection is bitopological in three meaningful ways. In particular it is shown that, apart from the assembly of a finitary biframe (mathcal {L}), there are two other structures (mathsf {A}_{cf}(mathcal {L})) and (mathsf {A}_{pm }(mathcal {L})), which both have the same main component as (mathsf {A}(mathcal {L})). The main component of both (mathsf {A}_{cf}(mathcal {L})) and (mathsf {A}_{pm }(mathcal {L})) is the ordered collection of all biquotients of (mathcal {L}). The structure (mathsf {A}_{cf}(mathcal {L})) being a biframe shows that the collection of all biquotients is generated by the frame of the patch-closed biquotients together with that of the patch-fitted ones. The structure (mathsf {A}_{pm }(mathcal {L})) being a biframe shows the collection of all biquotients is generated by the frame of the positive biquotients together with that of the negative ones. Notions of fitness and subfitness for finitary biframes are introduced, and it is shown that the analogues of two characterization theorems for these axioms hold. A spatial, bitopological version of these theorems is proven, in which finitary biframes whose spectrum is pairwise (T_1) are characterized, among other things in terms of the spectrum (mathsf {bpt}(mathsf {A}_{cf}(mathcal {L}))).

This paper concerns uniform continuity of real-valued functions on a (pre-)uniform frame. The aim of the paper is to characterize uniform continuity of such frame homomorphisms in terms of a farness relation between elements in the frame, and then to derive from it a separation and an extension theorem for real-valued uniform maps on uniform frames. The approach, purely order-theoretic, uses properties of the Galois maps associated with the farness relation. As a byproduct, we identify sufficient conditions under which a (continuous) scale in a frame with a preuniformity generates a real-valued uniform map.

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.