Optimal strong Mal’cev conditions for omitting type 1 in locally finite varieties

We show that the class of locally finite varieties omitting type 1 has the following properties. This class is:

1. definable by an idempotent, linear, strong Mal’cev condition in a language with one 4-ary function symbol;
2. not definable by an idempotent, linear, strong Mal’cev condition in a language with only one function symbol of arity strictly less than 4;
3. definable by an idempotent, linear, strong Mal’cev condition in a language with two 3-ary function symbols;
4. not definable by an idempotent, linear, strong Mal’cev condition in a language with function symbols of arity less than 4 unless at least two of the symbols have arity 3.

     

A strong Mal’cev condition for locally finite varieties omitting the unary type

We show that a finite algebra has a Taylor operation if and only if it has an operation satisfying a particular set of 6-ary Taylor identities. As a consequence we get the first strong Mal’cev condition for the family of locally finite varieties omitting the unary type. This is of interest to combinatorialists, as it is conjectured that a Constraint Satisfaction Problem defined by a core relational structure is polynomial time solvable exactly when a certain associated variety omits the unary type. Our result implies that the problem of deciding if a core relational structure meets this characterisation is itself in NP.     

Uniform Mal’cev algebras with small congruence lattices

A congruence of an algebra is called uniform if all the congruence classes are of the same size. An algebra is called uniform if each of its congruences is uniform. All algebras with a group reduct have this property. We prove that almost every finite uniform Mal’cev algebra with a congruence lattice of height at most two is polynomially equivalent to an expanded group.     

Strong Mal’tsev conditions in varieties of Abelian groups

Let Ab_d be a variety of Abelian groups of a finite exponent d≥1 and SC (Ab_d) be the set of all strong Mal’tsev conditions satisfied in Ab_d. We define the concept of a η-basis in SC(Ab_d) in terms of a basis w.r.t. a class η of varieties with commutative operations. The algorithm for constructing η-bases of any finite length in SC(Ab_d) is presented. For the variety Ab of all Abelian groups, we specify absolute bases of length 2 in SC(Ab) which are simultaneously η-bases. Bases of length 2 with similar properties are constructed also in SC(Ab_d), for any natural number d≥2. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 723–742, November–December, 1999.     

Polynomial clones of Mal’cev algebras with small congruence lattices

We show that the polynomials of every finite Mal’cev algebra with congruence lattice of height at most 2 can be described by a finite set of relations.     

2-Supernilpotent Mal’cev algebras

In this note we prove that a Mal’cev algebra is 2-supernilpotent ([1, 1, 1] =  0) if and only if it is polynomially equivalent to a special expanded group. This generalizes Gumm’s result that a Mal’cev algebra is abelian if and only if it is polynomially equivalent to a module over a ring.     

Congruence Distributivity in Goursat and Mal’cev Categories

We characterize the congruence distributive property for the Goursat and regular Malcev categories in terms of preservation of the intersection by direct images. This splits a previous characterization for the exact Malcev categories in two cases: plain congruence distributive property and weak congruence distributive property.Mathematics Subject Classifications (2000) Primary: 08B10, 08C05, 18C05; secondary: 08A30, 18D05.     

All congruence lattice identities implying modularity have Mal’tsev conditions

For an arbitrary lattice identity implying modularity (or at least congruence modularity) a Maltsev condition is given such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Maltsev condition.     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

Terms which are Mal’cev modulo some functions

We derive consequences from the existence of a term which satisfies Mal’cev identities (characterizing permutability) modulo two functions F and G from admissible relations to admissible relations. We also provide characterizations of varieties having a Mal’cev term modulo F and G. Received September 21, 2006; accepted in final form March 20, 2007.     

由~*-正则半群簇的Mal’cev积生成的簇

本文试图从泛代数的角度来研究*-正则半群．首先,我们定义了*-关系态射,讨论了它的性质;然后,用*-关系态射作为桥梁,刻划了二个*-正则半群簇的Mal’cev积生成的*-正则半群簇．特别地,我们确定了形如〈uog〉〈uoReB〉,〈uoReg〉的*-正则半群簇．     

Phragmén-Lindelöf conditions for graph varieties

For P_m ∈ ?[z₁, …, z_n], homogeneous of degree m we investigate when the graph of P_m in ?ⁿ⁺¹ satisfies the Phragmén-Lindelöf condition PL(?ⁿ⁺¹, log), or equivalently, when the operator $i{\partial \over \partial_{x_{n+1}}}+P_{m}(D)$ admits a continuous solution operator on C^∞(?ⁿ⁺¹). This is shown to happen if the varieties V_+- ? {z ∈ ?ⁿ: P_m(z) = ±1} satisfy the following Phragmén-Lindelöf condition (SPL): There exists A ≥ 1 such that each plurisubharmonic function u on V_+- satisfying u(z) ≤ ¦z¦+ o(¦z¦) on V_+- and u(x) ≤ 0 on V_+- ∩ ?ⁿ also satisfies u(z) A¦ Im z¦ on V_+-. Necessary as well as sufficient conditions for V_+- to satisfy (SPL) are derived and several examples are given.     

Local finiteness for Green’s relations in semigroup varieties

A semigroup variety is said to be locally 𝒦-finite, where 𝒦 stands for any of Green’s relations ?, ?, ?, 𝒟, or 𝒥, if every finitely generated semigroup in this variety has only finitely many 𝒦-classes. We characterize locally 𝒦-finite varieties of finite axiomatic rank in the language of “forbidden objects”.     

Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equations

Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several conditions on u ₀ to prove the existence of a unique strong solution with u(0) = u ₀ in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to prove the following optimal result with the weakest possible initial value condition and the largest possible solution class: Given u ₀, q, s as above and the Stokes operator A ₂, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions.