Algebra Universalis最新文献

In the late ‘90s, Banaschewski and Vermeulen showed that any localic group is complete in its two-sided uniformity. In this paper, we provide a method of constructing extensions of localic groups. The raison d’être of this paper, however, is to show using generators and relations that if the left and the right uniformity coincide, then the localic group must be complete.

A congruence system on an algebra (textbf{A}) is a tuple (langle theta _1,ldots ,theta _k,) (a_1,ldots ,a_krangle ) where (theta _1,ldots ,theta _k in mathop {textrm{Con}}textbf{A}), (a_1,ldots ,a_k in A) and (langle a_i,a_jrangle in theta _i vee theta _j) for all (i,j in {1,ldots ,k}). A solution to such a congruence system is an element (a in A) satisfying (langle a,a_irangle in theta _i) for all (i in {1,ldots ,k}). A tuple of congruences (langle theta _1,ldots , theta _krangle ) is said to be a Chinese Remainder tuple (CR tuple for short) of (textbf{A}) provided that every system (langle theta _1,ldots ,theta _k,a_1,ldots ,a_krangle ) with (a_1,ldots ,a_k in A) has a solution. Since two congruences (theta _1,theta _2) form a CR tuple if and only if they permute, the property of being a CR tuple is a generalization of the notion of permutability that makes sense for more than two congruences. The main result of this article is a characterization of CR tuples for finite algebras in dual discriminator varieties. As an application, we obtain a neat characterization of CR tuples for finite distributive lattices.

A poset ({mathbb {X}}) is said to be zigzag image-finite, if the least updownset (i.e., both an upset and a downset) containing x is finite, for all (xin X.) We show that a bi-Heyting algebra is profinite if and only if it is isomorphic to the lattice of upsets of a zigzag image-finite poset. Zigzag image-finite posets have the property of being disjoint unions of finite connected posets. Because of this, we equivalently show that a bi-Heyting algebra is profinite if and only if it is isomorphic to a direct product of simple finite bi-Heyting algebras.

Recently, the concepts of topoframe and topoboolean have been introduced as a generalization of point-free topology, and the relation between topobooleans and complete I-contact algebras (ICAs) has been studied. In this paper, we first introduce the ICA-topoboolean (B_{tau (C)}), in which (tau (C)) is induced from the complete ICA (B, C), and then characterize compact atomic ICA-topobooleans by their point clusters. As an example of the noncompact case, we determine all clusters of (big ( mathcal {P}(mathbb {R}), Cbig )), an ICA on the Boolean algebra of the power set of real numbers (mathbb {R}). Finally, we generalize the Smirnov compactification theorem from proximity spaces to atomic ICA-topobooleans.

The class of identical inclusions was defined by E. S. Lyapin. This is the class of universal formulas which is situated strictly between identities and universal positive formulas. Classes of semigroups defined by identical inclusions are called inclusive varieties. Inclusive varieties that cannot be defined by the first order formulas are called nonelementary inclusive varieties. We study nonelementary inclusive varieties of groups, Clifford semigroups and nilsemigroups. In particular, a criterion for an inclusive variety to be nonelementary is found and limit nonelementary inclusive varieties of abelian groups are described. We also describe the upper semilattice of nonelementary inclusive varieties of finite abelian groups and prove that it is uncountable. We find an uncountable set of nonelementary inclusive varieties of nilpotent class 3 and nil class 2 finite commutative semigroups and a limit nonelementary inclusive variety of nilsemigroups. We consider completely regular semigroups in semigroup signature with an additional unary operation and nilsemigroups in semigroup signature with the additional constant 0.

We initiate the study of the poset (mathcal{N}mathcal{O}(B)) of necessity operators on a boolean algebra B. We show that (mathcal{N}mathcal{O}(B)) is a meet-semilattice that need not be distributive. However, when B is complete, (mathcal{N}mathcal{O}(B)) is necessarily a frame, which is spatial iff B is atomic. In that case, (mathcal{N}mathcal{O}(B)) is a locally Stone frame. Dual results hold for the poset (mathcal{P}mathcal{O}(B)) of possibility operators. We also obtain similar results for the posets (mathcal {TNO}(B)) and (mathcal {TPO}(B)) of tense necessity and possibility operators on B. Our main tool is Jónsson-Tarski duality, by which such operators correspond to continuous and interior relations on the Stone space of B.

We show that the variety of monadic ortholattices is closed under MacNeille and canonical completions. In each case, the completion of L is obtained by forming an associated dual space X that is a monadic orthoframe. This is a set with an orthogonality relation and an additional binary relation satisfying certain conditions. For the MacNeille completion, X is formed from the non-zero elements of L, and for the canonical completion, X is formed from the proper filters of L. The corresponding completion of L is then obtained as the ortholattice of bi-orthogonally closed subsets of X with an additional operation defined through the binary relation of X. With the introduction of a suitable topology on an orthoframe, as was done by Goldblatt and Bimbó, we obtain a dual adjunction between the categories of monadic ortholattices and monadic orthospaces. A restriction of this dual adjunction provides a dual equivalence.

We give a definition of representability for distributive quasi relation algebras (DqRAs). These algebras are a generalisation of relation algebras and were first described by Galatos and Jipsen (Algebra Univers 69:1–21, 2013). Our definition uses a construction that starts with a poset. The algebra is concretely constructed as the lattice of upsets of a partially ordered equivalence relation. The key to defining the three negation-like unary operations is to impose certain symmetry requirements on the partial order. Our definition of representable distributive quasi relation algebras is easily seen to be a generalisation of the definition of representable relations algebras by Jónsson and Tarski (AMS 54:89, 1948). We give examples of representable DqRAs and give a necessary condition for an algebra to be finitely representable. We leave open the questions of whether every DqRA is representable, and also whether the class of representable DqRAs forms a variety. Moreover, our definition provides many other opportunities for investigations in the spirit of those carried out for representable relation algebras.

We study categorical properties of right-preordered groups, giving an explicit description of limits and colimits in this category, studying some exactness properties, and showing that it is a quasivariety. We show that, from an algebraic point of view, the category of right-preordered groups shares several properties with the one of monoids. Moreover, we describe split extensions of right-preordered groups, showing in particular that semidirect products of ordered groups always have a natural right-preorder.

Motivated by knot theory, it is natural to define the orienta-tion-reversal of a quandle orbit by inverting all the translations given by elements of that orbit. In this short note we observe that this natural notion is unsuited to medial quandles.