Algebra Universalis最新文献

We characterize several properties of core quandles in terms of the properties of their underlying groups. Specifically, we characterize connected cores providing an answer to an open question and present a standard homogeneous representation for them, which allows us to prove that simple core quandles are primitive.

The problem of determinability for free algebras in a given variety was posed by B. I. Plotkin in his lectures on universal algebraic geometry. This problem has been solved for free groups by E. Formanek, and for free semigroups and free monoids by G. Mashevitsky and B. M. Schein. We solve the determinability problem for free strict n-tuple semigroups as a natural generalization of free semigroups.

Call a finite relational structure k-Słupecki if its only surjective k-ary polymorphisms are essentially unary, and Słupecki if it is k-Słupecki for all (k ge 2). We present conditions, some necessary and some sufficient, for a reflexive digraph to be Słupecki. We prove that all digraphs that triangulate a 1-sphere are Słupecki, as are all the ordinal sums (m oplus n) ((m,n ge 2)). We prove that the posets (mathbb {P}= m oplus n oplus k) are not 3-Słupecki for (m,n,k ge 2), and prove there is a bound B(m, k) such that (mathbb {P}) is 2-Słupecki if and only if (n > B(m,k)+1); in particular there exist posets that are 2-Słupecki but not 3-Słupecki.

In 1966, Mal’cev proved that a class (mathcal {K}) of first-order structures with a specified signature is a quasivariety if and only if (mathcal {K}) contains a unit and is closed under isomorphic images, substructures, and reduced products. In this article, we present a proof of this theorem in (textsf{ZF}) (i.e., the Zermelo–Fraenkel set theory without the axiom of choice).

We introduce a notion of partial commutator in the context of a normal category with cokernels, as the commutator resulting from the recently introduced notion of partial commutativity. The partial commutator is defined by making slight modifications to the definition of the Huq commutator, and they agree in the unital context. We show that some fundamental properties of the Huq commutator can still be recovered. In particular, the partial commutator vanishes if and only if the morphisms partially commute, and in a normal category with finite colimits, it always exists and can be obtained via colimits as it is the case in absolute settings. As an application, we investigate partial commutators in the category (textsf{LAlg}) of L-algebras in the sense of W. Rump.

We start with a zero-dimensional frame L and an arbitrary integral domain A. We equip A with the discrete topology and consider the ring of A-valued continuous functions on L, which we denote by (A_dL). In this article, we classify both the classical ring of quotients and maximal ring of quotients of (A_dL), paying special attention to the case of ({mathfrak Z}L) the integer-valued continuous functions on L.

A semitopological group G is said to be an n-semitopological group, if for any (gin G) with (enot in overline{{g}}) there is a neighborhood W of e such that (gnot in W^{n}), where (nin mathbb {N}). The class of n-semitopological groups ((nge 2)) contains the class of paratopological groups and Hausdorff quasi-topological groups. Fix any (nin mathbb {N}). Properties of n-semitopological groups are studied, and questions about n-semitopological groups are posed. Some generalized metric properties of n-semitopological groups are discussed, which contains mainly results are that (1) each Hausdorff first-countable 2-semitopological group admits a coarser semi-metrizable topology; (2) each locally compact, Baire and (sigma )-compact 2-semitopological group is a topological group; (3) the condensation of some kind of 2-semitopological groups topologies are given. Finally, some cardinal invariants of n-semitopological groups are discussed.

Given an ideal I in the ring (mathcal {M}(X,mathcal {A})) of all real valued measurable functions over the measurable space ((X,mathcal {A})) and a measure (mu :mathcal {A}rightarrow [0,infty ]), we introduce the (u_mu ^I)-topology and the (m_mu ^I)-topology on (mathcal {M}(X,mathcal {A})) as generalizations of the u-topology and the m-topology on (mathcal {M}(X,mathcal {A})) respectively. For a countably generated ideal I in (mathcal {M}(X,mathcal {A})), it is proved that the (u_mu ^I)-topology and the (m_mu ^I)-topology coincide if and only if (Xsetminus bigcap Z[I]) is a (mu )-bounded subset of X. The components of 0 in both of these topologies are determined and it is proved that the condition of denseness of an ideal I in (mathcal {M}(X,mathcal {A})) is equivalent in these two topologies and this happens when and only when there exists (Zin Z[I]) such that (mu (Z)=0). It is also proved that I is closed in (mathcal {M}(X,mathcal {A})) in the (m_mu )-topology if and only if it is a (Z_mu )-ideal. Two more topologies on (mathcal {M}(X,mathcal {A})) viz. the (u_{mu ,F}^I)-topology and the (m_{mu ,F}^I)-topology, finer than the (u_mu ^I)-topology and the (m_mu ^I)-topology respectively are introduced and a few relevant properties are investigated thereon.

On a finite structure, the polymorphism invariant relations are exactly the primitively positively definable relations. On infinite structures, these two sets of relations are different in general. Infinitarily primitively positively definable relations are a natural intermediate concept which extends primitive positive definability by infinite conjunctions. We consider for every convex set (Ssubseteq {mathbb {R}}^n) the structure of the real numbers ({mathbb {R}}) with addition, scalar multiplication, constants, and additionally the relation S. We prove that depending on S, the set of all relations with an infinitary primitive positive definition in this structure equals one out of six possible sets. This dependency gives a natural partition of the convex sets into six nonempty classes. We also give an elementary geometric description of the classes and a description in terms of linear maps. The classification also implies that there is no locally closed clone between the clone of affine combinations and the clone of convex combinations.

Based on an analogue for systems of partial isomorphisms between lower sections in a complemented modular lattice we construct a series of terms (including inner inverse as basic operation and providing descending chains) such that principal right ideals (aR cong bR) in a (von Neumann) regular ring R are perspective if the series becomes stationary. In particular, this applies if (aR cap bR) is of finite height in L(R). This is used to derive, for existence-varieties (mathcal {V}) of regular rings, equivalence of unit-regularity and direct finiteness, both conceived as a property shared by all members of (mathcal {V}).