Connected CW Left Regular Bands

• Connected CW left regular bands are finite semigroups defined by idempotence and contraction properties that model face posets of regular CW complexes.
• They link algebra, combinatorial topology, and semigroup representation theory through structured support semilattices and regular CW complex intervals.
• Their algebraic invariants, quiver presentations, and connectivity criteria drive insights in representation theory, utilizing Möbius inversion and cellular topology.

A connected CW left regular band is a finite semigroup whose algebraic, combinatorial, and topological structures are deeply intertwined. Such bands serve as fundamental objects connecting regular CW complexes, combinatorial topology, and semigroup representation theory. This class encompasses the face semigroups of hyperplane arrangements, oriented matroids, and finite CAT(0) cube complexes. Rigorous framework and recent advances by Margolis–Saliola–Steinberg, Saliola, and others have clarified their structure, representations, and algebraic invariants (Margolis et al., 2015, Commins et al., 8 Dec 2025, Steinberg, 2024, Steinberg, 10 Jan 2026).

1. Definition and Structural Foundations

A left regular band (LRB) is a semigroup $B$ such that $x^2 = x$ and $xyx = xy$ for all $x, y \in B$. Each $B$ determines a support semilattice $\Lambda(B)$, the meet-semilattice of its principal left ideals $Bx$, ordered by inclusion. The surjective support map $\sigma : B \to \Lambda(B)$ satisfies $\sigma(xy) = \sigma(x) \wedge \sigma(y)$.

A CW left regular band (CW LRB) is an LRB $B$ such that for each $X \in \Lambda(B)$, the contraction $B_{\ge X} = \{b \in B : \sigma(b) \ge X\}$, ordered by the natural partial order $x \leq y$ if $yx = x$, is the face poset of a finite regular CW complex; thus, each closed interval has the topology of a ball, and each open interval is homeomorphic to a sphere. This confers Cohen–Macaulay and graded structure on $\Lambda(B)$. More generally, if $B = E(M)$ for a finite monoid $M$ in which each local group is abelian and $B$ is the face poset of a regular CW complex, then $M$ is called a CW left regular band of abelian groups (Steinberg, 2024, Margolis et al., 2015).

2. Connectivity Criteria

A CW LRB $B$ is called connected if each contraction $B_{\ge X}$ has a connected Hasse diagram (equivalently, the associated CW complex is topologically connected; the $1$-skeleton is a connected graph). Algebraically, $B$ is connected if and only if its semigroup algebra $kB$ over a field $k$ is unital. This is further equivalent to the property that each principal ideal $B_{\le y} = \{x: x \le y\}$ is connected in the Hasse diagram. These equivalences are foundational and are central to all subsequent representation-theoretic properties (Commins et al., 8 Dec 2025, Steinberg, 10 Jan 2026, Margolis et al., 2015):

• $B$ connected $\iff$ $kB$ is unital $\iff$ each $B_{\le y}$ is connected.

Connectivity of $B$ further implies the integral and modular representation theory and the quiver-theoretic structure are globally controlled by the connectivity of $\Lambda(B)$ and its intervals.

3. Topological Interpretation and Interval Structure

The poset structure $(B, \leq)$ defined by $x \leq y \iff yx = x$ yields a rich topological setting. Every principal ideal $B_{\le y}$ is the face poset of a regular CW ball, and each open interval $(p, q)$ is homeomorphic to a sphere of dimension $\operatorname{rk}(q) - \operatorname{rk}(p) - 1$ (Commins et al., 8 Dec 2025, Margolis et al., 2015). This provides a direct combinatorial correspondence between the algebraic structure of $B$ and the cellular topology of regular CW complexes. The support semilattice $\Lambda(B)$ inherits a graded, Cohen–Macaulay poset structure; the face posets of the contractions $B_{\ge X}$ serve as regular CW balls or spheres at each level.

For CW LRBs of abelian groups, the topology of the idempotent CW-complex $\Sigma$—with $B$ as its face poset—governs both the connectivity and the representation theory. Order complexes of lower (and upper) intervals control projective resolutions, Ext-groups, and quiver structure (Steinberg, 2024).

4. Representation Theory and Algebras

The semigroup algebra $kB$ of a connected CW LRB possesses a complete family of primitive orthogonal idempotents $\{e_X: X \in \Lambda(B)\}$ with $e_Xe_Y = \delta_{X,Y}e_X$ and $\sum_X e_X = 1$. The Peirce decomposition of $kB$ reads: $kB = \bigoplus_{X, Y \in \Lambda(B)} e_Y kB e_X$ The Cartan invariants are given by $$c_{X,Y} = \dim_k(e_Y kB e_X) = |\mu_{\Lambda}(X,Y)|$$, where $\mu_{\Lambda}$ denotes the Möbius function of $\Lambda(B)$.

The quiver of $kB$ (as a basic or path algebra modulo relations) has vertex set $\Lambda(B)$, and an arrow $X \to Y$ for each cover $X \prec Y$ in the semilattice. The defining relations in the path algebra are generated by the sum of all length-2 paths in each rank-2 interval: $$I = \left\langle \sum_{X < Z < Y} (X \to Z \to Y) \;\big|\; \operatorname{rk}[X, Y] = 2 \right\rangle$$ so $kB \cong kQ/I$ is quadratic and Koszul (Steinberg, 10 Jan 2026, Margolis et al., 2015, Steinberg, 2024). The Koszul dual is the incidence algebra of $\Lambda(B)^{op}$.

For left regular bands of abelian groups, the basic algebra's quiver vertices are pairs $(X, \chi)$ with $X \in \Lambda(B)$ and $\chi$ a character of the maximal abelian subgroup at $X$. Arrows and relations generalize the above via group character data.

5. Homological and Combinatorial Invariants

Ext-groups between simple modules for the semigroup algebra of a CW LRB are given explicitly by poset homology: $\Ext^n_{kB}(S_x, S_y) \cong H_{n-1}(\Delta(x, y); k)$ where $S_x, S_y$ are the simple modules corresponding to idempotent $x, y$, and $\Delta(x, y)$ is the order complex of the open interval $(x, y)$. The projective resolution of any simple module can be constructed via the cellular chain complex of the upper interval's CW ball.

Enumerative formulas for the sizes of fibers of the support map and other invariants are Möbius inversion expressions on $\Lambda(B)$: $|\sigma^{-1}(X)| = \sum_{Y \ge X} |\mu(Y, X)|$ These connect directly to face enumeration in associated geometric structures, such as the $f$-vector for CAT(0) cube complexes (Margolis et al., 2015).

6. Universal Quiver Presentations and Integral Models

For a connected CW LRB, the integral semigroup algebra $\mathbb{Z}B$ is isomorphic to the path algebra $\mathbb{Z}Q$ of the Hasse diagram $Q$ of $\Lambda(B)$ modulo the ideal generated by the sum of all length-2 paths: $$\mathbb{Z}B \cong \mathbb{Z}Q/I, \qquad I = \left\langle \sum_{\ell(p) = 2} p \right\rangle$$ where $p$ runs over all paths of length 2 in $Q$. This presentation holds over all fields by base change and is universal, for example recovering the face semigroup algebra of a hyperplane arrangement (Steinberg, 10 Jan 2026).

Worked examples, such as hyperplane faces in $\mathbb{R}^2$, demonstrate explicit isomorphisms and the combinatorial nature of these relations.

7. Examples and Group Actions

• Hyperplane arrangement face LRBs: $B = F(\mathcal{A})$ with $\Lambda(B)$ the intersection lattice, contractions as faces of zonotopes; $B$ is a CW LRB and is connected iff the arrangement is essential.
• CAT(0) cube complexes: $B$ is the face semigroup with $\Lambda(B)$ the hyperplane intersection semilattice; contractions are faces of convex subcomplexes, and connectivity reflects that of the complex.
• Oriented matroids (COMs): Covector monoids yield connected CW LRBs; contractions correspond to oriented-matroid contractions.
• Boolean flags: $B_n$ is not CW LRB, but its hereditary algebra retains much of the representation structure.

When a finite group $G$ acts by automorphisms on a CW LRB, $G$ permutes supports and idempotents, preserving connectivity and structure. The invariant subalgebra $(kB)^G$ is semisimple and commutative precisely when $|\Lambda(B)/G| = |B/G|$. Connectivity criteria then transfer to $G$-orbit contractions (Commins et al., 8 Dec 2025).

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The theory of connected CW left regular bands exemplifies a deep synthesis of combinatorial topology, algebraic combinatorics, and semigroup representation theory. Their role as bridges between poset topology, CW complexes, and the module theory of semigroups continues to inform current research directions and applications (Margolis et al., 2015, Commins et al., 8 Dec 2025, Steinberg, 2024, Steinberg, 10 Jan 2026).