Bicyclic Monoid: Algebraic & Combinatorial Analysis

• The bicyclic monoid is a prototypical non-group, bisimple inverse monoid defined by two generators with the relation pq = 1, serving as a minimal counterexample in semigroup theory.
• It exhibits a unique algebraic structure with explicit multiplication rules, descending idempotents, and clear Green's relations that inform its ideal and representation theory.
• Concrete models, including transformation semigroup and tropical matrix representations, highlight its practical applications in combinatorics, topology, and advanced algebra.

The bicyclic monoid is the prototypical example of a non-group, bisimple inverse monoid, appearing in numerous areas of algebra, semigroup theory, combinatorics, representation theory, and topological algebra. It is fundamental as the minimal counterexample to several group-like properties and as a universal object in the theory of one-relator semigroups.

1. Algebraic Structure and Presentation

The bicyclic monoid, typically denoted $\mathcal{B}$ or $B$, is the monoid generated by two elements $p$ and $q$ subject to the single defining relation

$pq = 1,$

where $1$ is the identity element. Every element can be represented uniquely in normal form as either $q^a p^b$ or $a^i b^j$ with $i, j \in \mathbb{N}^0$ (where the correspondence $a \leftrightarrow q$, $b \leftrightarrow p$ is standard in the literature).

The multiplication is explicitly given by

$$a^k b^\ell \cdot a^m b^n = \begin{cases} a^{k+m-\ell} b^n, & \ell \leq m \ a^k b^{\ell-m+n}, & \ell > m \end{cases}$$

or equivalently,

$$a^k b^\ell a^m b^n = a^{k-\ell+t} b^{n-m+t}, \quad t = \max\{\ell, m\}.$$

In this structure, $\mathcal{B}$ is an inverse monoid: each $a^i b^j$ has a unique inverse $a^j b^i$, and the idempotent elements are $e_n = a^n b^n$ for $n \geq 0$.

2. Green's Relations, Simplicity, and Idempotents

The Green's relations in $\mathcal{B}$ exhibit a trivial $\mathcal{H}$-class structure, with all $\mathcal{H}$-classes being singletons. The $\mathcal{R}$-classes are indexed by $i$ in the set $\{a^i b^j : j \in \mathbb{N}^0\}$, while the $\mathcal{L}$-classes are indexed by $j$, due to the form of the presentation.

There is exactly one $\mathcal{D}$-class, so $\mathcal{B}$ is bisimple and, in particular, simple. The idempotents $e_n = a^n b^n$ form a descending $\omega$-chain under the natural order ($e_0 = 1 \ge e_1 \ge \cdots$). Inverse semigroup theory applies in full generality, and the structure of the idempotents informs much of the representation and ideal theory of $\mathcal{B}$ (Ghroda, 2011, Ceccherini-Silberstein et al., 2013).

3. Categorical and Combinatorial Realizations

A concrete model of $\mathcal{B}$ involves viewing it as a submonoid of the transformation semigroup $\operatorname{Map}(\mathbb{N})$ under composition. Set

$$\begin{aligned} p(n) &= \begin{cases} n-1, & n \geq 1 \ 0, & n = 0 \end{cases}, \ q(n) &= n + 1. \end{aligned}$$

Then $p \circ q = \mathrm{Id}$ but $q \circ p \neq \mathrm{Id}$, demonstrating the essential non-cancellativity and lack of invertibility outside the group case (Ceccherini-Silberstein et al., 2013).

Combinatorially, $\mathcal{B}$ can be identified with $\mathbb{N}^0 \times \mathbb{N}^0$, with multiplication

$$(i, j) \cdot (k, \ell) = \begin{cases} (i + k - j, \ell), & j \leq k \ (i, \ell + j - k), & j > k \end{cases}$$

reflecting an infinite staircase structure in the Cayley graph.

4. Embeddings, Tropical Matrix Representation, and Identities

The bicyclic monoid admits a faithful representation in the semigroup of $2 \times 2$ upper-triangular tropical matrices over the tropical semiring $(\mathbb{R} \cup \{-\infty\}, \oplus, \otimes)$, where $\oplus$ is tropical addition (max) and $\otimes$ is tropical multiplication (addition) (Nyberg-Brodda, 2022). The embedding is given by: $$\begin{aligned} p &\mapsto B = \begin{pmatrix} 0 & 0 \ -\infty & -1 \end{pmatrix}, \ q &\mapsto A = \begin{pmatrix} 0 & 1 \ -\infty & 1 \end{pmatrix}, \end{aligned}$$ and

$$\rho(q^i p^j) = \begin{pmatrix} i - j & i + j \ -\infty & j - i \end{pmatrix}.$$

This embedding is a semigroup isomorphism onto its image.

A crucial theorem is that the semigroup identities satisfied by $\mathrm{UT}_2(\mathbb{T})$ are precisely the identities satisfied by the bicyclic monoid (Daviaud et al., 2016). For example, Adjan’s identity holds: $AB^2A^2BAB^2A = AB^2ABA^2B^2A.$ Nontrivial semigroup identities are algorithmically testable by the tropical polynomial method—see (Daviaud et al., 2016) for complexity results and explicit algorithms for identity checking.

Additionally, the free monogenic inverse semigroup and various further combinatorial inverse semigroups admit embeddings into upper-triangular tropical matrix semigroups (of possibly higher dimension), with the same identity theory as $\mathcal{B}$.

5. Soficity, Amenability, and Non-cancellativity

$\mathcal{B}$ is not left- or right-cancellative: for instance, $p(qp) = 1p = p$, yet $qp \neq 1$; similarly, $q(pq) = q1 = q$, but $pq \neq 1$. Nevertheless, $\mathcal{B}$ is an inverse monoid and is amenable (in the sense of possessing a finitely additive left invariant mean), as established in earlier work by Duncan and Namioka.

Importantly, the bicyclic monoid is not sofic (Ceccherini-Silberstein et al., 2013). In contrast to the group case (where no non-sofic group is yet known), $\mathcal{B}$ presents a finitely presented, amenable inverse monoid that is non-sofic—demonstrated by an obstruction based on the inability to approximate its dynamics by permutation actions on finite sets, a consequence of one-sided cancellation failure.

The proof shows that, for a specific finite subset $K = \{1, p, q, qp\}$ and small enough $\varepsilon > 0$, a putative nearly-multiplicative, nearly-injective map from $\mathcal{B}$ to a finite symmetric monoid cannot exist, as two different elements $qp \ne 1$ would become too close under the Hamming metric for a "sofic" approximation to be possible. This demonstrates subtleties not present in the group case.

6. Generalizations and Extensions

Several generalizations of the bicyclic monoid exist. The $\alpha$-bicyclic monoids $$\mathcal{B}_\alpha = \omega^\alpha \times \omega^\alpha$$ for ordinals $\alpha$ extend the classical structure, with a multiplication law mirroring the essential features of $\mathcal{B}$ but in a higher cardinality context (Bardyla, 2017).

Topologically, the only locally compact Hausdorff shift-continuous (semitopological) semigroup topology on the classical bicyclic monoid is discrete. For $\mathcal{B}_\alpha$, the lattice of such topologies is anti-isomorphic to the ordinal segment $[1, \alpha]$, with explicit base neighborhoods described (notably, the classical case corresponds to the Andersen–Eberhart–Selden theorem).

Abstractly, the monoid can be further extended to $$B_\omega^\mathscr{F} = \omega \times \omega \times \mathscr{F}$$, where $\mathscr{F}$ is an $\omega$-closed family of subsets of $\omega$. Under certain conditions (specifically, when $\mathscr{F}$ is a singleton of an inductive set), this extension is isomorphic to the classical bicyclic monoid. These generalizations subsume both the bicyclic monoid and the semigroup of $\omega \times \omega$ matrix units, and provide a categorical framework for inverse combinatorial semigroups (Gutik et al., 2021).

7. Orders, Quotients, and Structural Classifications

The theory of $I$-orders in inverse semigroups is particularly transparent in the case of the bicyclic monoid (Ghroda, 2011). Every subsemigroup $S \subseteq \mathcal{B}$ falls into one of the following types:

• Diagonal ($S \subseteq \{a^n b^n\}$).
• Upper ($S$ consists of all $a^i b^j$ with $j \geq i$).
• Lower ($i \geq j$).
• Two-sided (combinations/strips defined by index constraints).

Necessary and sufficient conditions are given for each type to be a left $I$-order, with the result that in all such cases the left $I$-order is straight (every $x$ can be written as $a^{-1}b$ with $a$ and $b$ in the same $\mathcal{R}$-class). The criteria hinge on divisibility and covering properties of indices, with divisibility "strips" (i.e., requiring $j - i$ to be divisible by $d$, and similar) being forbidden unless $d=1$. This fine combinatorial control illustrates the unique simplicity and rigidity of $I$-orders in $\mathcal{B}$, providing a key example in the theory of bisimple inverse $\omega$-semigroups.

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Key references for this exposition include Ghroda (Ghroda, 2011), Ceccherini-Silberstein and Coornaert (Ceccherini-Silberstein et al., 2013), Bardyla (Bardyla, 2017), Daviaud–Johnson–Kambites (Daviaud et al., 2016), Gutik–Mykhalenych (Gutik et al., 2021), and further developments in tropical algebra and one-relation monoids (Nyberg-Brodda, 2022).