Singularities of Toric Varieties Associated with Finite Distributive Lattices

With each finite lattice L we associate a projectively embedded scheme V(L); as Hibi has shown, the lattice D is distributive if and only if V(D) is irreducible, in which case it is a toric variety. We first apply Birkhoff's structure theorem for finite distributive lattices to show that the orbit decomposition of V(D) gives a lattice isomorphic to the lattice of contractions of the bounded poset of join-irreducibles of D. Then we describe the singular locus of V(D) by applying some general theory of toric varieties to the fan dual to the order polytope of P: V(D) is nonsingular along an orbit closure if and only if each fibre of the corresponding contraction is a tree. Finally, we examine the local rings and associated graded rings of orbit closures in V(D). This leads to a second (self-contained) proof that the singular locus is as described, and a similar combinatorial criterion for the normal link of an orbit closure to be irreducible.     

The semilattice tensor product of projective distributive lattices

Grant A. Fraser defined the semilattice tensor productA ⊗B of distributive latticesA, B and showed that it is a distributive lattice. He proved that ifA ⊗B is projective then so areA andB, that ifA andB are finite and projective thenA ⊗B is projective, and he gave two infinite projective distributive lattices whose semilattice tensor product is not projective. We extend these results by proving that ifA andB are distributive lattices with more than one element thenA ⊗B is projective if and only if bothA andB are projective and both have a greatest element. Presented by W. Taylor.     

Relation categories and coproduct congruence categories in universal algebra

It is known that a categoryV-Rel ofadmissible relations can be formed for any variety of algebrasV, such that morphismsAB correspond to subalgebras ofA x B. We adapt the relation category construction of Hilton and Wu to categoriesC with finite limits and colimits and an image factorization system. The existence ofC-Rel and a dualcograph constructionC-Cogr are proved equivalent to certain stability properties of pullbacks or pushouts forC. For algebraic varietiesV,V -Cogr exists iffV satisfies the amalgamation property (AP) and the congruence extension property (CEP). MorphismsAB inV-Cogr correspond to congruences on the coproductA + B. It is showed that congruence permutability (CP), the intersection property for amalgamations (IPA), the Hamiltonian property, and the property that congruences 6 are determined by the equivalence class [0] can be given characterizations in terms of interlocked pullbacks and pushouts in such a categoryC. A new property IDA (intersections determine amalgamations) is defined, which is dual to CP in this context. Familiar results, such as CP implies congruence modularity, can be proved in such categories. Dually, ifV satisfies AP, CEP, IPA and IDA, it has modular lattices of subalgebras. These results are related to order duality for Su and Con. (For certain varietiesV, the subalgebras ofA are in one-one correspondence with the morphisms below 1_A inV -Rel orV-Cogr, and the congruences correspond to the morphisms above 1_A.) IfV is pointed (eachA in V has a smallest trivial subalgebra), then a category formulation is obtained for: CP implies the Jónsson-Tarski decomposition properties. The dual shows that pointed varieties satisfying IDA have a restricted form, with pointed unary varieties and varieties ofR-modules as special cases.Dedicated to Bjarni Jónsson on his 70th birthday ntprbPresented by G. McNulty.     

On a strong property of the weak subalgebra lattice

In the present paper, we apply results from [Pió1] to prove that for an arbitrary total and locally finite unary algebra A of finite unary type K, its weak subalgebra lattice uniquely determines its strong subalgebra lattice (recall that in the case of total algebras the strong subalgebra lattice is the well-known lattice of all (total) subalgebras). More precisely, we prove that for every unary partial algebra B of the same unary type K, if weak subalgebra lattices of A and B are isomorphic (with A as above), then the strong subalgebra lattices of A and B are isomorphic, and moreover B is also total and locally finite. At the end of this paper we also show the necessity of all the three conditions for A. Received September 5, 1997; accepted in final form October 7, 1998.     

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.     

The M3[D] construction and n-modularity

In 1968, Schmidt introduced the M ₃[D] construction, an extension of the five-element modular nondistributive lattice M ₃ by a bounded distributive lattice D, defined as the lattice of all triples satisfying . The lattice M ₃[D] is a modular congruence-preserving extension of D.? In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity such that is modularity and is properly weaker than . Let M _n denote the variety defined by , the variety of n-modular lattices. If L is n-modular, then M ₃[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, Id M ₃[L] M ₃[Id L]. ? We provide an example of a lattice L such that M ₃[L] is not a lattice. This example also provides a negative solution to a problem of Quackenbush: Is the tensor product of two lattices A and B with zero always a lattice. We complement this result by generalizing the M ₃[L] construction to an M ₄[L] construction. This yields, in particular, a bounded modular lattice L such that M ₄ L is not a lattice, thus providing a negative solution to Quackenbush’s problem in the variety M of modular lattices.? Finally, we sharpen a result of Dilworth: Every finite distributive lattice can be represented as the congruence lattice of a finite 3-modular lattice. We do this by verifying that a construction of Gr?tzer, Lakser, and Schmidt yields a 3-modular lattice. Received May 26, 1998; accepted in final form October 7, 1998.     

Ideal extensions of lattices

Following the well-known Schreier extension of groups, the (ideal) extension of semigroups (without order) have been first considered by A. H. Clifford in Trans. Amer. Math. Soc. 68 (1950), with a detailed exposition of the theory in the monographs of Clifford-Preston and Petrich. The main theorem of the ideal extensions of ordered semigroups has been considered by Kehayopulu and Tsingelis in Comm. Algebra 31 (2003). It is natural to examine the same problem for lattices. Following the ideal extensions of ordered semigroups, in this paper we give the main theorem of the ideal extensions of lattices. Exactly as in the case of semigroups (ordered semigroups), we approach the problem using translations. We start with a lattice L and a lattice K having a least element, and construct (all) the lattices V which have an ideal L′ which is isomorphic to L and the Rees quotient V|L′ is isomorphic to K. Conversely, we prove that each lattice which is an extension of L by K can be so constructed. An illustrative example is given at the end. The text was submitted by the author in English.     

M-solid varieties generated by lattices

Following W. Taylor, we define an identity to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices. Received October 8, 1999; accepted in final form March 22, 2000.     

A test for identities satisfied in lattices of submodules

SupposeR is ring with 1, andH?(R) denotes the variety of modular lattices generated by the class of lattices of submodules of allR-modules. An algorithm using Mal'cev conditions is given for constructing integersm≧0 andn≧1 from any given lattice polynomial inclusion formulad∈e. The main result is thatd∈e is satisfied in every lattice inH?(R) if and only if there existsx inR such that (m·1)x=n·1 inR, where 0·1=0 andk·1=1+1...+1 (k times) fork≧1. For example, this “divisibility” condition holds form=2 andn=1 if and only if 1+1 is an invertible element ofR, and it holds form=0 andn=12 if and only if the characteristic ofR divides 12. This result leads to a complete classification of the lattice varietiesH?(R),R a ring with 1. A set of representative rings is constructed, such that for each ringR there is a unique representative ringS satisfyingH?(R)=H?(R). There is exactly one representative ring with characteristick for eachk≧1, and there are continuously many representative rings with characteristic zero. IfR has nonzero characteristic, then all free lattices inH?(R) have recursively solvable word problems. A necessary and sufficient condition onR is given for all free lattices inH?(R) to have recursively solvable word problems, ifR is a ring with characteristic zero. All lattice varieties of the formH?(R) are self-dual. A varietyH?(R) is a congruence variety, that is, it is generated by the class of congruence lattices of all members of some variety of algebras. A family of continuously many congruence varieties related to the varietiesH?(R) is constructed.     

On the number of join-irreducibles in a congruence representation of a finite distributive lattice

For a finite lattice L, let $ \trianglelefteq_L $ denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form $ \trianglelefteq_L $, as follows: Theorem. Let $ \trianglelefteq $ be a quasi-ordering on a finite set P. Then the following conditions are equivalent:(i) There exists a finite lattice L such that $ \langle J(L), \trianglelefteq_L $ is isomorphic to the quasi-ordered set $ \langle P, \trianglelefteq \rangle $.(ii) $ |\{x\in P|p \trianglelefteq x\}| \neq 2 $, for any $ p \in P $.For a finite lattice L, let $ \mathrm{je}(L) = |J(L)|-|J(\mathrm{Con} L)| $ where Con L is the congruence lattice of L. It is well-known that the inequality $ \mathrm{je}(L) \geq 0 $ holds. For a finite distributive lattice D, let us define the join- excess function:$ \mathrm{JE}(D) =\mathrm{min(je} (L) | \mathrm{Con} L \cong D). $We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that $ \mathrm{JE}(D) \leq (2/3)| \mathrm{J}(D)|$ , for any finite distributive lattice D; the constant 2/3 is best possible.A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.Dedicated to the memory of Gian-Carlo Rota     

Monotone path systems in simple regions

A monotone path system (MPS) is a finite set of pairwise disjoint paths (polygonal areas) in thexy-plane such that every horizontal line intersects each of the paths in at most one point. A MPS naturally determines a pairing of its top points with its bottom points. We consider a simple polygon in thexy-plane wich bounds the simple polygonal (closed) regionD. LetT andB be two finite, disjoint, equicardinal sets of points ofD. We give a good characterization for the existence of a MPS inD which pairsT withB, and a good algorithm for finding such a MPS, and we solve the problem of finding all MPSs inD which pairT withB. We also give sufficient conditions for any such pairing to be the same.The first author's research is supported by the Natural Sciences and Engineering Research Council of Canada     

Construction of hjelmslev planes from (t,k)-nets

A (t, k)-net is an abstract generalization of the incidence structures which occur as the point and line neighborhoods of a finite Hjelmslev plane. A (t, k)-net contains ‘substructures’ which are nets of ordert and degreek. Every (t, r) Hjelmslev plane (brieflyH-plane) can be constructed from a suitable collection of (t, r+1)- and (t, r)-nets. A (t, r)H-plane or (t, k)-net is called extremal provided: each two points which are joined by more than two lines are joined by preciselyt lines and dually. IfB is a ‘properly’ extremal (t, r)H-plane (means both 2 andt≠2 occur among the joining numbers), thent is even; andr=2 orr=1+(t/2). All the 3-uniform [J. Combinat. Theory 9, 267–288 (1970, this Zbl.204, 210)] (4, 2)H-planes are examples. Two further examples are constructed in the paper: an (8, 2) translationH-planeC and an (8, 2) projectiveH-planeD, none of whose affineH-planes are translationH-planes. All point neighborhoods fromC andD and all line neighborhoods fromD are isomorphic to a give (8, 3)-netE;E is constructed by considering the subspaces of a 64-point symplectic geometry overZ ₂.E is also used to answer (affirmatively) the question of the existence of proper fairly near affineH-planes [J. Combinat. Theory 16A, 34–50 (1974)].     

Inequalities for convex bodies and polar reciprocal lattices inR n

LetL be a lattice and letU be ano-symmetric convex body inR ⁿ . The Minkowski functional ∥ ∥ _U ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i (L,U)i=1,...,n, are defined in the usual way. Let ℒ _n be the family of all lattices inR ⁿ . Given a pairU,V of convex bodies, we define and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ _n andu∈R ⁿ /(L+U), somev∈L ^* with ∥v∥ _V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U ⁰), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl _p ⁿ , 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U ⁰) are less thanCn logn for some numerical constantC.     

Radical classes of algebras with B-action

We initiate the radical theory of algebras with B-action where B is a fixed Boolean ring. We consider lattices of classes of algebras defined in terms of ideals of B. In two special cases (universal classes of -groups with B-action and idempotent algebras with B-action), these ideal-defined classes are sublattices of the lattice of radicals, and we characterise semisimplicity in such cases. Received February 2, 1998; accepted in final form June 11, 1998.     

Commensurability invariants for nonuniform tree lattices

We study nonuniform lattices in the automorphism groupG of a locally finite simplicial treeX. In particular, we are interested in classifying lattices up to commensurability inG. We introduce two new commensurability invariants:quotient growth, which measures the growth of the noncompact quotient of the lattice; andstabilizer growth, which measures the growth of the orders of finite stabilizers in a fundamental domain as a function of distance from a fixed basepoint. WhenX is the biregular treeX _m,n, we construct lattices realizing all triples of covolume, quotient growth, and stabilizer growth satisfying some mild conditions. In particular, for each positive real numberν we construct uncountably many noncommensurable lattices with covolumeν. Supported in part by NSF grants DMS-9704640 and DMS-0244542. Supported in part by an NSF postdoctoral research fellowship.     

Catalytic distributive lattices and compact zero-dimensional topological lattices

The (Priestley) dual spaces ofD ₀₁-catalytic lattices are analysed and shown to be precisely the compact zero-dimensional topological lattices. This characterisation is used to prove that a bounded distributive lattice isD ₀₁-catalytic if and only if it is a retract of one freely generated by an ordered set.Presented by I. Rival.     

On finite retract lattices of monounary algebras

For a monounary algebra (A, f) we denote R ^∅(A, f) the system of all retracts (together with the empty set) of (A, f) ordered by inclusion. This system forms a lattice. We prove that if (A, f) is a connected monounary algebra and R ^∅(A, f) is finite, then this lattice contains no diamond. Next distributivity of R ^∅(A, f) is studied. We find a representation of a certain class of finite distributive lattices as retract lattices of monounary algebras.     

One Setting for All: Metric, Topology, Uniformity, Approach Structure

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T,V)-algebras and of (T,V)-proalgebras turn out to be topological over Set.     

Isometries of JB-algebras

A bijective linear mapping between two JB-algebrasA andB is an isometry if and only if it commutes with the Jordan triple products ofA andB. Other algebraic characterizations of isometries between JB-algebras are given. Derivations on a JB-algebraA are those bounded linear operators onA with zero numerical range. For JB-algebras of selfadjoint operators we have: IfH andK are left Hilbert spaces of dimension ≥3 over the same fieldF (the real, complex, or quaternion numbers), then every surjective real linear isometryf fromS(H) ontoS(K) is of the formf(x)=UoxoU ⁻¹ forx inS(H), whereτ is a real-linear automorphism ofF andU is a real linear isometry fromH ontoK withU(λh)=τ(λ)U(h) for λ inF andh inH. Supported by Acción Integrada Hispano-Alemana HA 94 066 B