On some non-obvious connections between graphs and unary partial algebras

In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras A and B, their weak subalgebra lattices are isomorphic if and only if their graphs G*(A) and G*(B) are isomorphic. Secondly, it is shown that for two unary partial algebras A and B if their digraphs G(A) and G(B) are isomorphic, then their (weak, relative, strong) subalgebra lattices are also isomorphic. Thirdly, we characterize pairs , where A is a unary partial algebra and L is a lattice such that the weak subalgebra lattice of A is isomorphic to L.     

On the Subalgebra Lattice of Unary Algebras

We characterize pairs L, A, where Lis a lattice and Ais a unary partial algebra, such that the strong subalgebra lattice S_s(A) is isomorphic to L. Moreover, we find necessary and sufficient conditions for arbitrary unary partial algebras to have isomorphic strong subalgebra lattices. Observe, that for a total algebra A, the lattice S_s(A) is the usual well-known subalgebra lattice. Thus in particular we solve these two problems for total unary algebras and their lattices of (also total) subalgebras.For this purpose we apply some non-obvious connections between unary partial algebras and graphs from [9]. More precisely, we first characterize the pairs L, G, where Lis a lattice and Ga directed graph, such that the strong subdigraph lattice of Gis isomorphic to L. Next, we find a characterization of arbitrary digraphs with isomorphic strong subalgebra lattices. From these results we easily get solutions of our algebraic problems.     

The subalgebra lattice of a finite algebra

The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.     

The Homomorphism Lattice Induced by a Finite Algebra

Each finite algebra A induces a lattice L _A via the quasi-order → on the finite members of the variety generated by A, where B →C if there exists a homomorphism from B to C. In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice L _A induced by a finite algebra A?’ Our main result is that each finite distributive lattice arises as L _Q , for some quasi-primal algebra Q. We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form L ⊕1, where L is an interval in the subgroup lattice of a finite group.     

Congruence-preserving subdirect products

An algebra A is said to be a congruence-preserving extension of a subalgebra B if the mapping from the congruence lattice of B to that of A, assigning to each congruence relation β on B the minimal congruence relation on A containing β, is an isomorphism. We give a necessary and sufficient condition on the congruence lattice of a subdirect product B of finitely many algebras in a congruence-distributive variety that the full direct product be a congruence-preserving extension of B. We give several applications to congruence lattices of lattices. Received May 25, 2000; accepted in final form January 22, 2001.     

A uniform Birkhoff theorem

Birkhoff’s HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra B satisfies all equations that hold in an algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a (possibly infinite) direct power of A. The former statement is equivalent to the existence of a natural map sending term functions of the algebra A to those of B—the natural clone homomorphism. The study of continuity properties of natural clone homomorphisms has been initiated recently by Bodirsky and Pinsker for locally oligomorphic algebras.Revisiting the argument of Bodirsky and Pinsker, we show that for any algebra B in the variety generated by an algebra A, the induced natural clone homomorphism is uniformly continuous if and only if every finitely generated subalgebra of B is a homomorphic image of a subalgebra of a finite power of A. Based on this observation, we study the question as to when Cauchy continuity of natural clone homomorphisms implies uniform continuity. We introduce the class of almost locally finite algebras, which encompasses all locally oligomorphic as well as all locally finite algebras, and show that, in case A is almost locally finite, then the considered natural homomorphism is uniformly continuous if (and only if) it is Cauchy-continuous. In particular, this provides a locally finite counterpart of the result by Bodirsky and Pinsker. Along the way, we also discuss some peculiarities of oligomorphic permutation groups on uncountable sets.     

Embeddings into Orthomodular Lattices with Given Centers,State Spaces and Automorphism Groups

We prove that, given a nontrivial Boolean algebra B, a compact convex set S and a group G, there is an orthomodular lattice L with the center isomorphic to B, the automorphism group isomorphic to G, and the state space affinely homeomorphic to S. Moreover, given an orthomodular lattice J admitting at least one state, L can be chosen such that J is its subalgebra.     

Almost distributive sublattices and congruence heredity

A congruence lattice L of an algebra A is hereditary if every 0-1 sublattice of L is the congruence lattice of an algebra on A. Suppose that L is a finite lattice obtained from a distributive lattice by doubling a convex subset. We prove that every congruence lattice of a finite algebra isomorphic to L is hereditary. Presented by E. W. Kiss. Received July 18, 2005; accepted in final form April 2, 2006.     

An analog of McKenzie’s theorem for topology lattices of finite algebras

In this paper, it is shown that the topology lattice of any finite algebra is isomorphic to the topology lattice of some finite algebra with four unary operations. Further, we present countably many unary algebras whose topology lattices are distributive and nonisomorphic to a topology lattice of any unar (a unar is an algebra with one unary operation). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 109–117, 2005.     

On quasiorder lattices and topology lattices of algebras

In this paper, it is shown that the dual [(\textQord)\tilde]\mathfrakA \widetilde{\text{Qord}}\mathfrak{A} of the quasiorder lattice of any algebra \mathfrakA \mathfrak{A} is isomorphic to a sublattice of the topology lattice á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . Further, if \mathfrakA \mathfrak{A} is a finite algebra, then [(\textQord)\tilde]\mathfrakA @ á( \mathfrakA ) \widetilde{\text{Qord}}\mathfrak{A} \cong \Im \left( \mathfrak{A} \right) . We give a sufficient condition for the lattices [(\textCon)\tilde]\mathfrakA\text, [(\textQord)\tilde]\mathfrakA \widetilde{\text{Con}}\mathfrak{A}{\text{,}} \widetilde{\text{Qord}}\mathfrak{A} , and á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras.     

Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.     

Gelfand–Kirillov Dimension of Algebras with a Locally Finite and Multiparameter Indexed Filtration

Let A have a locally finite and multiparameter indexed filtration ?, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ?. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.     

Solutions to five problems on tensor products of lattices and related matters

The notion of a capped tensor product, introduced by G. Grätzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extend many results from the finite case to the infinite case. In this paper, we answer several open questions about tensor products of lattices. Among the results that we obtain are the following:¶¶Theorem 2. Let A be a lattice with zero. If A ?B A \oplus B is a lattice for every lattice L with zero, then A is locally finite and A ?B A \oplus B is a capped tensor product for every lattice L with zero.¶¶Theorem 5. There exists an infinite, three-generated, 2-modular lattice K with zero such that K ?K K \oplus K is a capped tensor product.¶¶Here, 2-modularity is a weaker identity than modularity, introduced earlier by G. Grätzer and the author.     

Table algebras with multiple P-polynomial structures

Using covering numbers we prove that a standard real integral table algebra (A, B) with |B| ≥ 6 has a P-polynomial structure with respect to every b ≠ 1 in B if and only if 2|B|-1 is prime and (A, B) is exactly isomorphic to the Bose-Mesner algebra of the association scheme of the ordinary (2|B|-1)-gon. Then we present an example showing that this result is not true if |B| ≤ 5.     

Finitely Deep Matrices

In the algebraic study of deep matrices ? _X () on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal  whose quotient is a central simple algebra. He showed that this ideal, which doesn't appear for infinite index sets, is itself a central simple algebra. In this article we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of ? _X (A)/ and  are isomorphic to the lattice of ideals of the coordinate algebra A.     

Subalgebra extensions of partial monounary algebras

For a subalgebra B of a partial monounary algebra A we define the quotient partial monounary algebra A/B. Let B, B, C be partial monounary algebras. In this paper we give a construction of all partial monounary algebras A such that B is a subalgebra of A and C ≅ A/B.     

On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim(A + [A, L])/A < . We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A <. We show that the condition of local finiteness of L is essential in this statement.     

Generalized Graded Hecke Algebras of Types B and D

For a root system of type B we study an algebra similar to a graded Hecke algebra, isomorphic to a subalgebra of the rational Cherednik algebra. We introduce principal series modules over it and prove an irreducibility criterion for these modules. We deduce similar results for an algebra associated to a root system of type D.     

Semigroups that are factors of subdirectly irreducible semigroups by their monolith

Jeek and Kepka [4] proved that a universal algebra A with at least one at least binary operation is isomorphic to the factor of a subdirectly irreducible algebra B by its monolith if and only if the intersection of all of its (nonempty) ideals is nonempty, and that B may be chosen to be finite if A is finite. (By an ideal of A is meant a non-empty subset I of A such that f(a₁, . . . , a_n) I whenever f is an n-ary fundamental operation of A and a₁, . . . , a_n A are elements with a_i I for at least one index i.) In the present paper, we prove that if A is a semigroup, then B may be chosen also to be a semigroup, but that a finite semigroup need not be isomorphic to the factor of a finite subdirectly irreducible semigroup by its monolith.