Relative separation in distributive congruence lattices

In [5] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in Now we generalize these results using the concept of relatively separable sets (with respect to subsets) and apply them to some lattice varieties.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived November 29, 2002; accepted in final form August 19, 2004.     

Inflation of Finite Lattices Along All-or-nothing Sets

We introduce a new generalization of Alan Day’s doubling construction. For ordered sets \(\mathcal {L}\) and \(\mathcal {K}\) and a subset \(E \subseteq \ \leq _{\mathcal {L}}\) we define the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) arising from inflation of \(\mathcal {L}\) along E by \(\mathcal {K}\). Under the restriction that \(\mathcal {L}\) and \(\mathcal {K}\) are finite lattices, we find those subsets \(E \subseteq \ \leq _{\mathcal {L}}\) such that the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) is a lattice. Finite lattices that can be constructed in this way are classified in terms of their congruence lattices.A finite lattice is binary cut-through codable if and only if there exists a 0?1 spanning chain \(\left \{\theta _{i}\colon 0 \leq i \leq n \right \}\) in \(Con(\mathcal {L})\) such that the cardinality of the largest block of ?? _i /?? _i?1 is 2 for every i with 1≤i≤n. These are exactly the lattices that can be constructed by inflation from the 1-element lattice using only the 2-element lattice. We investigate the structure of binary cut-through codable lattices and describe an infinite class of lattices that generate binary cut-through codable varieties.     

Joins and subdirect products of varieties

We generalise in three different directions two well-known results in universal algebra. Grätzer, Lakser and P?onka proved that independent subvarieties \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) of a variety \({\mathcal{V}}\) are disjoint and such that their join \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) (in the lattice of subvarieties of \({\mathcal{V}}\)) is their direct product \({\mathcal{V}_{1} \times \mathcal{V}_{2}}\) . Jónsson and Tsinakis provided a partial converse to this result: if \({\mathcal{V}}\) is congruence permutable and \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) are disjoint, then they are independent (and so \({\mathcal{V}_{1} \vee \mathcal{V}_{2} = \mathcal{V}_{1} \times \mathcal{V}_{2}}\)). We show that (i) if \({\mathcal{V}}\) is subtractive, then Jónsson’s and Tsinakis’ result holds under some minimal assumptions; (ii) if \({\mathcal{V}}\) satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) is the subdirect product of \({\mathcal{V}_{1}}\) and \({\mathcal{V}_2}\) ; (iii) the same holds if \({\mathcal{V}}\) is congruence 3-permutable.     

Meet-irreducible congruence lattices

The system of all congruences of an algebra (A, F) forms a lattice, denoted \({{\mathrm{Con}}}(A, F)\). Further, the system of all congruence lattices of all algebras with the base set A forms a lattice \(\mathcal {E}_A\). We deal with meet-irreducibility in \(\mathcal {E}_A\) for a given finite set A. All meet-irreducible elements of \(\mathcal {E}_A\) are congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were described in Jakubíková-Studenovská et al. (2017). In this paper, we prove necessary and sufficient conditions under which \({{\mathrm{Con}}}(A, f)\) is meet-irreducible in the case when (A, f) is an algebra with short tails (i.e., f(x) is cyclic for each \(x \in A\)) and in the case when (A, f) is an algebra with small cycles (every cycle contains at most two elements).     

Semilattices of Definable Subalgebras

In issues bearing on the structure of universal algebras , derived structures, such as automorphism groups Aut , subalgebra lattices Sub , congruence lattices Con , etc., play an important part. On the other hand, in studying universal algebras by the means of model theory, of crucial importance is the question asking which elements of the derived structures under examination are expressible by one or other formulas in the elementary language. Problems concerning the interrelationship of algebras and their derived structures are treated for subalgebras of universal algebras.Supported by RFBR grant No. 02-01-00258.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 474–482, July–August, 2005.     

Compact factor congruences imply Boolean factor congruences

We prove that any variety in which every factor congruence is compact has Boolean factor congruences, i.e., for all A in the set of factor congruences of A is a distributive sublattice of the congruence lattice of A.     

Congruence properties in congruence permutable and in ideal determined varieties, with applications.

We define a weak version of EDPC (equationally definable principal congruences), called EDPC*, that is shown to be preserved under varietal closure in congruence permutable varieties. We show that if is a congruence permutable variety generated by a class then has EDPC iff has EDPC* iff has EDPC*. An equational condition is given which, if satisfied by implies that has the CEP (congruence extension property). Similar results are proved for ideal determined varieties. These results are applied to the variety of residuated lattices, with examples.Received January 15, 2004; accepted in final form October 8, 2004.     

Isotone maps on lattices

Let ${\mathcal{L} = (Li | i \in I)}$ be a family of lattices in a nontrivial lattice variety V, and let ${\varphi_{i} : L_{i} \rightarrow M}$ , for ${i \in I}$ , be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the maps ${\varphi_{i}}$ can be extended to an isotone map ${\varphi : L \rightarrow M}$ , where ${L = {\rm Free}_{V} \mathcal{L}}$ is the free product of the L _i in V. This was known for V = L, the variety of all lattices. The above free product L can be viewed as the free lattice in V on the partial lattice P formed by the disjoint union of the L _i . The analog of the above result does not, however, hold for the free lattice L on an arbitrary partial lattice P. We show that the only codomain lattices M for which that more general statement holds are the complete lattices. On the other hand, we prove the analog of our main result for a class of partial lattices P that are not-quite-disjoint unions of lattices. We also obtain some results similar to our main one, but with the relationship lattices : orders replaced either by semilattices : orders or by lattices : semilattices. Some open questions are noted.     

On the variety of almost commuting nilpotent matrices

Let V be an n-dimensional vector space over an algebraically closed field and $\mathcal{N}Let V be an n-dimensional vector space over an algebraically closed field and N\mathcal{N} the nilcone of nilpotent endomorphisms of V. We study the variety A = {(X, Y, i, j) ? N ×N ×V ×V^* |[X, Y] = ij} \mathcal{A} = \left\{{(X, Y, i, j) \in \mathcal{N} \times \mathcal{N} \times V \times V^{\ast} \vert [X, Y] = ij}\right\} which is closely related to the variety of pairs of nilpotent n × n matrices with commutator of rank at most 1. We describe its irreducible components: two of them correspond to the pairs of commuting matrices, and n − 2 components of smaller dimension corresponding to the pairs of rank one commutator. In our proof we define a map to the zero fiber of the Hilbert scheme of points and study the image and the fibers.     

Imaginaries in Boolean algebras

Given an infinite Boolean algebra B, we find a natural class of $\varnothing$‐definable equivalence relations $\mathcal {E}_{B}$ such that every imaginary element from B^eq is interdefinable with an element from a sort determined by some equivalence relation from $\mathcal {E}_{B}$. It follows that B together with the family of sorts determined by $\mathcal {E}_{B}$ admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of imaginaries for Boolean algebras, obtained in 10 .     

The Ground State and the Long-Time Evolution in the CMC Einstein Flow

Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)³Vol_g(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by V_inf=(\frac-16Y(S))^\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g _i , K _i )} satisfying: (i) k _i  = −3, (ii) V_iˉ V_inf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q ₀((g _i , K _i )) ≤ Λ, where Q ₀ is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)²g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E₁)\tilde]=E₁(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E₁=Q₀+Q₁{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ V_inf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.     

A generalization of Mathieu subspaces to modules of associative algebras

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $ \mathcal{A} $ \mathcal{A} , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} , where R is the base ring of $ \mathcal{A} $ \mathcal{A} . We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.     

Varieties generated by modes of submodes

In a natural way, we can ??lift?? any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra ( ${A, \Omega}$ ) its power algebra of subsets. G. Gr?tzer and H. Lakser proved that for a variety ${\mathcal{V}}$ , the variety ${\mathcal{V}\Sigma}$ generated by power algebras of algebras in ${\mathcal{V}}$ satisfies precisely the consequences of the linear identities true in ${\mathcal{V}}$ . For certain types of algebras, the sets of their subalgebras form subalgebras of their power algebras. They are called the algebras of subalgebras. In this paper, we partially solve a long-standing problem concerning identities satisfied by the variety ${\mathcal{VS}}$ generated by algebras of subalgebras of algebras in a given variety ${\mathcal{V}}$ . We prove that if a variety ${\mathcal{V}}$ is idempotent and entropic and the variety ${\mathcal{V}\Sigma}$ is locally finite, then the variety ${\mathcal{VS}}$ is defined by the idempotent and linear identities true in ${\mathcal{V}}$ .     

Locally finite varieties of Heyting algebras

We show that for a variety of Heyting algebras the following conditions are equivalent: (1) is locally finite; (2) the -coproduct of any two finite -algebras is finite; (3) either coincides with the variety of Boolean algebras or finite -copowers of the three element chain are finite. We also show that a variety of Heyting algebras is generated by its finite members if, and only if, is generated by a locally finite -algebra. Finally, to the two existing criteria for varieties of Heyting algebras to be finitely generated we add the following one: is finitely generated if, and only if, is residually finite. Received November 11, 2001; accepted in final form July 25, 2005.     

The Cycle Discrepancy of Three-Regular Graphs

Let G = (V, E) be an undirected graph and C(G){{\mathcal C}(G)} denote the set of all cycles in G. We introduce a graph invariant cycle discrepancy, which we define as

Applications of Operator Space Theory to Nest Algebra Bimodules

Recently, Blecher and Kashyap have generalized the notion of W ^*-modules over von Neumann algebras to the setting where the operator algebras are σ closed algebras of operators on a Hilbert space. They call these modules weak* rigged modules. We characterize the weak* rigged modules over nest algebras. We prove that Y is a right weak* rigged module over a nest algebra Alg(M){\rm{Alg}(\mathcal M)} if and only if there exists a completely isometric normal representation F{\Phi } of Y and a nest algebra Alg(N){\rm{Alg}(\mathcal N)} such that Alg(N) F(Y)Alg(M) ì F(Y){\rm{Alg}(\mathcal N) \Phi (Y)\rm{Alg}(\mathcal M)\subset \Phi (Y)} while F(Y){\Phi (Y)} is implemented by a continuous nest homomorphism from M{\mathcal M} onto N{\mathcal N} . We describe some properties which are preserved by continuous CSL homomorphisms.     

Traces, traceability, and lattices of traces under the set theoretic inclusion

Let a trace be a computably enumerable set of natural numbers such that ${V^{[m]} = \{n : \langle n, m\rangle \in V \}}$ V [ m ] = { n : 〈 n , m 〉 ∈ V } is finite for all m, where ${\langle^{.},^{.}\rangle}$ 〈 . , . 〉 denotes an appropriate pairing function. After looking at some basic properties of traces like that there is no uniform enumeration of all traces, we prove varied results on traceability and variants thereof, where a function ${f : \mathbb{N} \rightarrow \mathbb{N}}$ f : N → N is traceable via a trace V if for all ${m, \langle f(m), m\rangle \in V.}$ m , 〈 f ( m ) , m 〉 ∈ V . Then we turn to lattices $$\textit{\textbf{L}}_{tr}(V) = (\{W : V \subseteq W \, {\rm and} \, W \, {\rm a} \, {\rm trace}\}, \, \subseteq),$$ L t r ( V ) = ( { W : V ? W and W a trace } , ? ) , V a trace. Here, we study the close relationship to ${\mathcal{E} = (\{A : A \subseteq \mathbb{N} \quad c.e.\}, \subseteq)}$ E = ( { A : A ? N c . e . } , ? ) , automorphisms, isomorphisms, and isomorphic embeddings.     

Series-parallel posets and relative Ockham lattices

A study is undertaken of order-reversing maps on series-parallel posets and structural characterisations are obtained of various subclasses of such ordered sets. The results are applied to complete the authors' earlier investigation of classes of finite relate lattices, where is a variety of Ockham lattices and the distributive lattice duals of the algebras in are required to be series-parallel.     

On the Complexity Functions for T-Ideals of Associative Algebras

Let be the sequence of codimension growth for a variety V of associative algebras. We study the complexity function , which is the exponential generating function for the sequence of codimensions. Earlier, the complexity functions were used to study varieties of Lie algebras. The objective of the note is to start the systematic investigation of complexity functions in the associative case. These functions turn out to be a useful tool to study the growth of varieties over a field of arbitrary characteristic. In the present note, the Schreier formula for the complexity functions of one-sided ideals of a free associative algebra is found. This formula is applied to the study of products of T-ideals. An exact formula is obtained for the complexity function of the variety U c of associative algebras generated by the algebra of upper triangular matrices, and it is proved that the function is a quasi-polynomial. The complexity functions for proper identities are investigated. The results for the complexity functions are applied to study the asymptotics of codimension growth. Analogies between the complexity functions of varieties and the Hilbert--Poincaré series of finitely generated algebras are traced.     

Congruences and ideals on Boolean modules: a heterogeneous point of view

Definitions for heterogeneous congruences and heterogeneous ideals on a Boolean module $\mathcal {M}$ are given and the respective lattices $\mathrm{Cong}\mathcal {M}$ and $\mathrm{Ide}\mathcal {M}$ are presented. A characterization of the simple bijective Boolean modules is achieved differing from that given by Brink in a homogeneous approach. We construct the smallest and the greatest modular congruence having the same Boolean part. The same is established for modular ideals. The notions of kernel of a modular congruence and the congruence induced by a modular ideal are introduced to describe an isomorphism between $\mathrm{Cong}\mathcal {M}$ and $\mathrm{Ide}\mathcal {M}$. This isomorphism leads us to conclude that the class of the Boolean module is ideal determined.