Amalgamation and Injectivity in Banach Lattices

We study distinguished objects in the category $\mathcal{B}\mathcal{L}$ of Banach lattices and lattice homomorphisms. The free Banach lattice construction introduced by de Pagter and Wickstead [ 8] generates push-outs, and combining this with an old result of Kellerer [ 17] on marginal measures, the amalgamation property of Banach lattices is established. This will be the key tool to prove that $L_1([0,1]^{\mathfrak{c}})$ is separably $\mathcal{B}\mathcal{L}$-injective, as well as to give more abstract examples of Banach lattices of universal disposition for separable sublattices. Finally, an analysis of the ideals on $C(\Delta ,L_1)$, which is a separably universal Banach lattice as shown by Leung et al. [ 21], allows us to conclude that separably $\mathcal{B}\mathcal{L}$-injective Banach lattices are necessarily non-separable.

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

On a class of -modules

UDC 512.5Smith in paper [<em>Mapping between module lattices,</em> Int. Electron. J. Algebra, <strong>15</strong>, 173–195 (2014)] introduced maps between the lattice of ideals of a commutative ring and the lattice of submodules of an -module i.e., and mappings.The definitions of the maps were motivated by the definition of multiplication modules.Moreover, some sufficient conditions for the maps to be a lattice homomorphisms are studied.In this work we define a class of -modules and observe the properties of the class. We give a sufficient conditions for the module and the ring such that the class is a hereditary pretorsion class.

Bounded homomorphisms and finitely generated fiber products of lattices

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean’s condition (D) which arises from Dean’s solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman’s condition satisfy (D). For lattice epimorphisms [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are finitely generated and [Formula: see text] satisfies (D), we show the following: If [Formula: see text] and [Formula: see text] are bounded, then their fiber product (pullback) [Formula: see text] is finitely generated. While the converse is not true in general, it does hold when [Formula: see text] and [Formula: see text] are free. As a consequence, we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

Lattice Homomorphisms Between Weak Orders

We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly, surjective homomorphisms exist quite generally: They exist if and only if the diagram of $W'$ is obtained from the diagram of $W$ by deleting vertices, deleting edges, and/or decreasing edge labels. A surjective homomorphism $W\to W'$ is determined by its restrictions to rank-two standard parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter groups, this determination in rank two is nontrivial. Indeed, from the combinatorial lattice theory point of view, all of these classification results should appear unlikely a priori. As an application of the classification of surjective homomorphisms between weak orders, we also obtain a classification of surjective homomorphisms between Cambrian lattices and a general construction of refinement relations between Cambrian fans.

Primary Decomposition of Ideals of Lattice Homomorphisms

For two given finite lattices $L$ and $M$, we introduce the ideal of lattice homomorphism $J(L,M)$, whose minimal monomial generators correspond to lattice homomorphisms $\phi : L\to M$. We show that $L$ is a distributive lattice if and only if the equidimensinal part of $J(L,M)$ is the same as the equidimensional part of the ideal of poset homomorphisms $I(L,M)$. Next, we study the minimal primary decomposition of $J(L,M)$ when $L$ is a distributive lattice and $M=[2]$. We present some methods to check if a monomial prime ideal belongs to $\mathrm{ass}(J(L,[2]))$, and we give an upper bound in terms of combinatorial properties of $L$ for the height of the minimal primes. We also show that if each minimal prime ideal of $J(L,[2])$ has height at most three, then $L$ is a planar lattice and $\mathrm{width}(L)\leq 2$. Finally, we compute the minimal primary decomposition when $L=[m]\times [n]$ and $M=[2]$.

‘POSITIVELY HOMOGENEOUS LATTICE HOMOMORPHISMS BETWEEN RIESZ SPACES NEED NOT BE LINEAR’

This note furnishes an example showing that the main result (Theorem 4) in Toumi [‘When lattice homomorphisms of Archimedean vector lattices are Riesz homomorphisms’, J. Aust. Math. Soc. 87 (2009), 263–273] is false.