Background. Let $n$ be a nonnegative integer, and let $S_n$ denote the $n$th symmetric group. The $0$Hecke monoid $H_0left(S_nright)$ is defined to be the monoid given by generators $t_1, t_2, ldots, t_{n1}$ and relations

$t_i^2 = t_i$ for every $i in left{1, 2, ldots, n1right}$;

$t_i t_{i+1} t_i = t_{i+1} t_i t_{i+1}$ for every $i in left{1, 2, ldots, n2right}$;

$t_i t_j = t_j t_i$ for every $i$ and $j$ in $left{1, 2, ldots, n1right}$ satisfying $leftijright > 1$.
(This is a particular case of a more general construction — that of the $0$Hecke monoid of a Coxeter group –; but I want to focus on the symmetric group.)
The monoid $H_0left(S_nright)$ has cardinality $n!$. More precisely, for every permutation $w in S_n$, we can define an element $t_w$ of $H_0left(S_nright)$ by setting
$t_w = t_{i_1} t_{i_2} cdots t_{i_k}$, where $left(i_1, i_2, ldots, i_kright)$ is a reduced word for $w$ in $S_n$.
This element $t_w$ does not depend on the choice of the reduced word. The family $left(t_wright)_{w in S_n}$ contains each element of $H_0left(S_nright)$ exactly once. Thus, we can define a monoid structure $left(S_n, *right)$ on the set $S_n$ as follows: For any $a in S_n$ and $b in S_n$, define $a * b$ to be the unique element of $S_n$ satisfying $t_{a * b} = t_a t_b$. Then, $left(S_n, *right)$ is an isomorphic copy of the $0$Hecke monoid $H_0left(S_nright)$ whose elements are those of $S_n$. It has various interesting properties, which are spread across the literature (apparently very popular as exercises).
Question. Is there an equivalent definition of $*$ that is “synthetic”, i.e., does not use the decomposition of a permutation into adjacent transpositions? (My intuition for $a * b$ is something along the lines of “the join of $a$ and $b$ on the Bruhat order lattice, if one squints hard enough to forget that the Bruhat order is not a lattice and that $*$ is not commutative”.)
Motivation. For comparison, here is a similar object where the answer is “Yes”. We define a zeroed monoid to be a monoid $M$ with a specified element $0$ which satisfies $0m = m0 = 0$ for every $m in M$. We can define the nilCoxeter monoid $C_0left(S_nright)$ to be the monoid given by generators $u_1, u_2, ldots, u_{n1}, 0$ and relations

$0 u_i = u_i 0 = 0$ for every $i in left{1, 2, ldots, n1right}$;

$u_i^2 = 0$ for every $i in left{1, 2, ldots, n1right}$;

$u_i u_{i+1} u_i = u_{i+1} u_i u_{i+1}$ for every $i in left{1, 2, ldots, n2right}$;

$u_i u_j = u_j u_i$ for every $i$ and $j$ in $left{1, 2, ldots, n1right}$ satisfying $leftijright > 1$.
Similar results as for $H_0left(S_nright)$ hold; in particular, we can define a $u_w in C_0left(S_nright)$ for every $w in S_n$, and the family $left(u_wright)_{w in S_n}$ contains each element of $C_0left(S_nright) setminus left{0right}$ exactly once. We can again transfer this zeroed monoid structure onto the set $S_n cup left{0right}$; in other words, we can define a binary operation $sharp$ on $S_n cup left{0right}$ by $u_{a sharp b} = u_a u_b$ for all $a, b in S_n cup left{0right}$ (where I set $u_0 = 0$). But this multiplication $sharp$ can be described without using reduced words: Namely, for any $a in S_n$ and $b in S_n$, we have
$a sharp b = begin{cases} ab, & text{ if } ellleft(abright) = ellleft(aright) + ellleft(bright); \ 0, & text{ otherwise} end{cases}$,
where $ellleft(wright)$ means the Coxeter length of a permutation of $w$ (that is, the number of inversions of $w$). Of course, the Coxeter length of a permutation $w$ is the length of any reduced expression of $w$, but it can also be defined as the number of inversions of $w$, which does not rely on the representation of a permutation as composition of adjacent transpositions. I am looking for a similar description for the $0$Hecke monoid.
(Note: None of the results above is mine, but it is hard to find proofs for them in the literature, so I don’t even know whom to cite.)
Comment converted to an answer on OP’s suggestion.
This mathoverflow question: Is the following construction of the 0Hecke monoid (well) known? might be relevant to your question.
Let me also mention two possibly relevant references
On the representation theory of finite Jtrivial monoids by
Tom Denton, Florent Hivert, Anne Schilling, Nicolas M. Thiéry
A Combinatorial Formula for Orthogonal Idempotents in the $0$Hecke Algebra of the Symmetric Group by
Tom Denton
Monoid H_0(S_n) can be defined as a monoid of permutation matrices under a special kind of multiplication, or equivalently of special integer matrices (unitMonge matrices) under standard tropical multiplication. See my paper for details: https://doi.org/10.1007/s004530139830z
I am curious about your remark that the monoid’s properties are popular as exercises: do you have any examples of those?