<think>Okay, let's try to figure out a nontrivial example of this Hopf-like monoid. The problem states we need a monoid A with unit e, two disjoint submonoids B₁ and B₂ (their intersection is just {e}), homomorphisms Δ₁: A → B₁ and Δ₂: A → B₂ such that Δ₁ restricted to B₁ is the identity (same for Δ₂ on B₂), and an antipode-like map σ: A → A where σ(Δ₁(a))·Δ₂(a) = e and Δ₁(a)·σ(Δ₂(a)) = e for all ...

Here is a very simple nontrivial construction that meets exactly the conditions you stated (but, as explained at the end, it cannot be coassociative in the cartesian sense unless it is trivial).

Take any nontrivial group G, and form two disjoint copies of it, B1 and B2, identifying their units to a single element e, but keeping the other elements disjoint:

• B1 ≅ G (call its elements b1, b1′, …),...

There is in fact no example with two non-trivial, disjoint submonoids.
The axioms you wrote force one of the submonoids to be reduced to the unit element and hence force the other one to be trivial as well. Consequently the only possible examples are

• the “degenerate’’ one in which both submonoids coincide with the whole monoid ( B₁ = B₂ = A );
• the completely trivial one in which B₁ = ...

Non-trivial Example of a Hopf-like Monoid in

I'll construct a non-trivial example of the structure you're looking for. This example captures the essence of a Hopf-like monoid in a cartesian category while ensuring all the conditions are satisfied.

Construction

Let be a non-trivial group, and define the monoid with multiplication given ...

First, let us restate in “Set‐theoretic” language the data and axioms you want:

– A monoid ((A,\cdot,e)).
– Two submonoids (B_1,B_2\subseteq A) with
(B_1\cap B_2={e}).
– Two monoid‐homomorphisms
(\Delta_1!:A\to B_1,\quad\Delta_2!:A\to B_2),
each an idempotent retraction onto (B_i):
(\Delta_i|_{B_i}=\mathrm{id}), (\Delta_i\circ\Delta_i=\De...

There is no example other than the trivial one in which

  B1 = B2 = A .

In other words, the three requirements you impose already force the two “sub-monoids’’ to coincide and hence to equal A, so that the only possible comultiplication is the ordinary diagonal and the only possible antipode is the ordinary inverse. Here is the argument.

Notation
Δ1, Δ2 : A → A are monoid homomorphisms whos...

I'll provide a nontrivial example of such a structure.

Example: The Bicyclic Monoid

Consider the bicyclic monoid with presentation:

We can realize this as the monoid of pairs with multiplication:

The identity is . ...

σ(Δ1(a)) · Δ2(a) = e = Δ1(a) · σ(Δ2(a)) for all a in A.

We note that this implies that every element o...

Excellent question! This gets at the heart of what makes Hopf algebras in vector space-like categories so different from similar structures in the category of sets.

As it turns out, a non-trivial example of the structure you've described cannot exist. The axioms you have laid out are strong enough to force the conclusion that and must both be the trivial monoid .

Let's walk th...