---
title: 'Bicyclic Monoid: Algebraic & Combinatorial Analysis'
url: https://www.emergentmind.com/topics/bicyclic-monoid
type: topic
---

# Bicyclic Monoid: Algebraic & Combinatorial Analysis

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 $\mathcal{B}$ [1107.3303], [1304.4919].

## 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 [1304.4919].

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) [2209.12612]. 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 [1612.04219]. 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 [1612.04219] 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 [1304.4919]. 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 [1707.07130]. 

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 [2107.14118].

## 7. Orders, Quotients, and Structural Classifications

The theory of $I$-orders in inverse semigroups is particularly transparent in the case of the bicyclic monoid [1107.3303]. 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 [1107.3303], Ceccherini-Silberstein and Coornaert [1304.4919], Bardyla [1707.07130], Daviaud–Johnson–Kambites [1612.04219], Gutik–Mykhalenych [2107.14118], and further developments in tropical algebra and one-relation monoids [2209.12612].

Source: https://www.emergentmind.com/topics/bicyclic-monoid