We introduce semidistrim lattices, a simultaneous generalization of semidistributive and trim lattices that preserves many of their common properties. We prove that the elements of a semidistrim lattice correspond to the independent sets in an associated graph called the Galois graph, that products and intervals of semidistrim lattices are semidistrim and that the order complex of a semidistrim lattice is either contractible or homotopy equivalent to a sphere.

Semidistrim lattices have a natural rowmotion operator, which simultaneously generalizes Barnard’s $\overline \kappa $ map on semidistributive lattices as well as Thomas and the second author’s rowmotion on trim lattices. Every lattice has an associated pop-stack sorting operator that sends an element x to the meet of the elements covered by x. For semidistrim lattices, we are able to derive several intimate connections between rowmotion and pop-stack sorting, one of which involves independent dominating sets of the Galois graph.

A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution {ck}. Notoriously hard to compute, the exact fractions ck have been determined for k ≤ 3 only. We present a shortcut enabling us to compute c4 and c5 as well; both are ratios of enormous integers, the denominator of c5 being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of ck is at most 2k+1 + 1. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.

Generalizing earlier results of Katriňák, El-Assar and the present author we prove new structure theorems for l-algebras. We obtain necessary and sufficient conditions for the decomposition of an arbitrary bounded lattice into a direct product of (finitely) subdirectly irreducible lattices.

In general, the tensor product, $A\otimes B$, of the lattices $A$ and $B$ with zero is not a lattice (it is only a join-semilattice with zero). If $A\otimes B$ is a capped tensor product, then $A\otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices $A$ with zero enjoying the property that $A\otimes B$ is a capped tensor product, for every lattice $B$ with zero; we shall call such lattices amenable.

The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition $\left( \text{T} \right)$.

We prove that a finite lattice $A$ is amenable iff it is sharply transferable as a join-semilattice.

For a general lattice $A$ with zero, we obtain the result: $A$ is amenable iff $A$ is locally finite and every finite sublattice of $A$ is transferable as a join-semilattice.

This yields, for example, that a finite lattice $A$ is amenable iff $A\otimes \text{F}\left( 3 \right)$ is a lattice iff $A$ satisfies $\left( \text{T} \right)$, with respect to join. In particular, ${{M}_{3}}\,\otimes \,\text{F}\left( 3 \right)$ is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

We consider certain pseudovarieties K of lattices which are closed under the doubling of convex sets. For each such K, given an arbitrary finite lattice 𝓛, we describe the covers of the variety V(𝓛) of the form V(𝓛, K) with K a subdirectly irreducible lattice in K.

This paper examines the simultaneous doubling of multiple intervals of a lattice in great detail. In the case of a finite set of W-failure intervals, it is shown that there in a unique smallest lattice mapping homomorphically onto the original lattice, in which the set of W-failures is removed. A nice description of this new lattice is given. This technique is used to show that every lattice that is a bounded homomorphic image of a free lattice has a projective cover. It is also used to give a sufficient condition for a fintely presented lattice to be weakly atomic and shows that the problem of which finitely presented lattices are finite is closely related to the problem of characterizing those finite lattices with a finite W-cover.