Completely Join-Irreducible Elements

• Completely join-irreducible elements are the minimal building blocks in a lattice, defined by the existence of a unique largest lower bound.
• They facilitate canonical decompositions and efficient computations in algorithms within representation theory and combinatorial settings.
• Applications span spatial lattice embeddings, automata in Coxeter groups, and insights into algebraic and geometric structures.

A completely join-irreducible element, often called a “point,” plays a fundamental role in lattice theory and associated combinatorics, representation theory, and geometric group theory. In a join-semilattice (in particular, a lattice) $L$, an element $p$ is completely join-irreducible if the set of strict lower bounds of $p$ has a largest element. Equivalently, $p$ cannot be expressed as the join of any proper collection of strictly smaller elements; any such join must include $p$ itself. This property governs the minimal “building blocks” from which all other elements of the lattice can be constructed by joins. The structure, density, and behavior of these elements strongly influence the geometry and algebraic properties of the lattice, the varieties they generate, and algorithms in combinatorics and representation theory.

1. Formal Definitions and Characterizations

Let $L$ be a (complete) lattice. An element $j \in L$ is completely join-irreducible if there exists a unique $j_* < j$ such that every $x < j$ satisfies $x \leq j_*$; equivalently,

$$\{ x \in L : x < j \} = \{ x \in L : x \leq j_* \}.$$

In terms of joins,

$j = \bigvee X \implies j \in X$

for any subset $X \subseteq L$.

Completely join-irreducible elements contrast with join-irreducible elements in that the former concerns arbitrary (possibly infinite) joins, not just finite joins. The set of all completely join-irreducible elements of $L$ is typically denoted $\Jc(L)$ or $\cji(L)$.

Dually, an element $m\neq \top$ in $L$ is completely meet-irreducible if it is covered by exactly one element and is not the meet of a collection omitting $m$.

In compactly generated lattices, every completely join-irreducible element is compact; in algebraic atomistic lattices, atoms coincide with the set of points, which always form a seed (see below) (Santocanale et al., 2011).

2. Structural Role in Lattices

Completely join-irreducible elements serve as the “atomic” components for decomposition in join-semilattices and lattices:

• Spatial lattices: A lattice $L$ is spatial if every element is a (possibly infinite) join of points; i.e., $\Jc(L)$ is join-dense,

$\forall a \in L,\quad a = \bigvee \{ p \in \Jc(L): p \leq a \}.$

• Strong spatiality via seeds: To impose further structure, especially over minimal join covers, one introduces the notion of a “seed”: a subset $\Sigma \subseteq L$ is a seed if it is contained in the join-irreducibles, is join-dense, and every finite cover of its element can be refined to a minimal join-cover by elements in $\Sigma$. $L$ is strongly spatial if $\Jc(L)$ is a seed. Every strongly spatial lattice is spatial, but the converse can fail outside distributive or modular lattices (Santocanale et al., 2011).

In Coxeter-theoretic contexts, completely join-irreducible elements characterize transitions in automata and minimality in right weak and Bruhat orderings, capturing critical combinatorial and geometric structure (Yau, 29 Oct 2024, Ko et al., 2021).

3. Canonical Decompositions and the κ-map

A central tool in the paper of semidistributive lattices is the canonical join-representation. For a semidistributive lattice $L$:

• Every element $x$ can be written as

$x = \bigvee_{j \in \mathrm{CJR}(x)} j$

for a unique minimal set of completely join-irreducible elements ($\mathrm{CJR}(x)$) (Barnard et al., 2022, Barnard et al., 2019).

• There exists a bijection, the κ-map (or rowmotion),

$\kappa: \cji(L) \to \cmi(L)$

mapping each completely join-irreducible to its associated completely meet-irreducible, with explicit lattice-theoretic constructions. This bijection underpins the combinatorics of canonical decompositions, flag simplicial complexes defined by canonical join relations, and, in representation theory, connects with objects such as bricks and Auslander-Reiten translations (Barnard et al., 2022, Barnard et al., 2019).

In semidistributive lattices arising from torsion classes of Artin algebras, completely join-irreducible torsion classes correspond bijectively to bricks. The κ-map translates to taking orthogonals to bricks, which forms the backbone of recent studies on reductions and exceptional sequences (Barnard et al., 2019).

4. Lattice Embeddings and Varieties

Joining properties of completely join-irreducible elements drive embedding results:

• Algebraic, strongly spatial hulls: For any $n$-distributive lattice $L$, there exists an embedding, within its variety, into an algebraic, strongly spatial lattice. The construction proceeds via seed generation and takes the ideal lattice over the semilattice generated by the points, ensuring that the embedding preserves varietal and (when present) zero/one elements (Santocanale et al., 2011).
• In the modular setting, every modular lattice can be analogously embedded; spatiality alone suffices to imply strong spatiality (Santocanale et al., 2011).

The existence (or failure) of such embeddings directly affects the structure of the corresponding varieties: for distributive varieties, word problems are decidable by finite generation; for modular varieties, such properties may fail (Santocanale et al., 2011).

5. Combinatorial Realizations in Coxeter and Weyl Groups

In Coxeter groups and Bruhat orders, completely join-irreducible elements admit uniform, type-dependent geometric/combinatorial descriptions:

• In right weak Coxeter order, tight gates (elements with a unique final root) are the completely join-irreducible elements, serving as fundamental “gates” in automata that recognize reduced words. Their explicit classification involves root combinatorics and minimality conditions in various classes (dihedral, right-angled, complete graph, rank-3 Coxeter types) (Yau, 29 Oct 2024).
• In strong Bruhat order, completely join-irreducible elements correspond to bigrassmannian elements (exactly one left and one right descent). In type A, there is a complete identification; in other types, classifications rely on chain- or family-structured subsets. These elements underpin the unique minimal join-expressions and have deep connections to the representation theory of Verma modules and Kazhdan–Lusztig polynomials (Ko et al., 2021).
• The canonical decompositions into join-irreducible elements reflect the geometry of Schubert varieties, automata minimization, and algebraic data such as socle series in category $\mathcal{O}$.

6. Nonexistence and Limit Phenomena

Not all lattices admit nontrivial completely join-irreducible elements:

• In the “continuous weak order” $L(I^d)$, despite a rich structure of (finite) join-irreducibles constructed via join-continuous functions, no element is completely join-irreducible. Every candidate element is the supremum of an infinite strictly increasing chain of irreducibles and thus fails the total primality condition. Dually, there are no nontrivial compact elements in these lattices (Gouveia et al., 2018).
• This phenomenon illustrates starkly the difference between “discrete” and “continuous” lattice-theoretic regimes: in the former, complete join-irreducibility frequently governs algebraic generation and spatiality; in the latter, infinite decomposability can defeat these properties.

7. Applications, Algorithms, and Decidability

Completely join-irreducible elements underpin both theoretical algorithms and practical computation:

• In automata theory for Coxeter groups, efficient algorithms exist to identify tight gates—completely join-irreducible elements—allowing rapid recognition of cone types and avoidance of full automaton computation (Yau, 29 Oct 2024).
• In variety theory, the join-density and seed properties enable reductions to finite models, yielding decidability results for word problems in distributive lattice varieties (Santocanale et al., 2011).
• In representation theory, canonical join-representations and the κ-map provide structural insights into the lattice of torsion classes and its interplay with bricks, wide subcategories, and the Auslander–Reiten translation (Barnard et al., 2022, Barnard et al., 2019).

Topic Area	Primary Source(s)	arXiv ID
Definition and spatial lattices	Wehrung et al.	(Santocanale et al., 2011)
Semidistributivity and κ-map	Barnard-Hanson, Barnard et al.	(Barnard et al., 2022, Barnard et al., 2019)
Coxeter groups and automata	Przytycki-Yau et al.	(Yau, 29 Oct 2024)
Bruhat order, Verma modules	Elian et al.	(Ko et al., 2021)
Lack in continuous order	Gouveia, Santocanale	(Gouveia et al., 2018)

Completely join-irreducible elements thus are the critical granularity underlying spatial decompositions, canonical representations, and computational reductions across a diverse range of modern algebraic and combinatorial contexts.