Background. Let $n$ be a nonnegative integer, and let $S_n$ denote the $n$-th symmetric group. The $0$-Hecke monoid $H_0\left(S_n\right)$ is defined to be the monoid given by generators $t_1, t_2, \ldots, t_{n-1}$ and relations
$t_i^2 = t_i$ for every $i \in \left\{1, 2, \ldots, n-1\right\}$;
$t_i t_{i+1} t_i = t_{i+1} t_i t_{i+1}$ for every $i \in \left\{1, 2, \ldots, n-2\right\}$;
$t_i t_j = t_j t_i$ for every $i$ and $j$ in $\left\{1, 2, \ldots, n-1\right\}$ satisfying $\left|i-j\right| > 1$.
(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 $H_0\left(S_n\right)$ has cardinality $n!$. More precisely, for every permutation $w \in S_n$, we can define an element $t_w$ of $H_0\left(S_n\right)$ by setting
$t_w = t_{i_1} t_{i_2} \cdots t_{i_k}$, where $\left(i_1, i_2, \ldots, i_k\right)$ 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 $\left(t_w\right)_{w \in S_n}$ contains each element of $H_0\left(S_n\right)$ 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 $H_0\left(S_n\right)$ 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 $C_0\left(S_n\right)$ to be the monoid given by generators $u_1, u_2, \ldots, u_{n-1}, 0$ and relations
$0 u_i = u_i 0 = 0$ for every $i \in \left\{1, 2, \ldots, n-1\right\}$;
$u_i^2 = 0$ for every $i \in \left\{1, 2, \ldots, n-1\right\}$;
$u_i u_{i+1} u_i = u_{i+1} u_i u_{i+1}$ for every $i \in \left\{1, 2, \ldots, n-2\right\}$;
$u_i u_j = u_j u_i$ for every $i$ and $j$ in $\left\{1, 2, \ldots, n-1\right\}$ satisfying $\left|i-j\right| > 1$.
Similar results as for $H_0\left(S_n\right)$ hold; in particular, we can define a $u_w \in C_0\left(S_n\right)$ for every $w \in S_n$, and the family $\left(u_w\right)_{w \in S_n}$ contains each element of $C_0\left(S_n\right) \setminus \left\{0\right\}$ exactly once. We can again transfer this zeroed monoid structure onto the set $S_n \cup \left\{0\right\}$; in other words, we can define a binary operation $\sharp$ on $S_n \cup \left\{0\right\}$ by $u_{a \sharp b} = u_a u_b$ for all $a, b \in S_n \cup \left\{0\right\}$ (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
$a \sharp b = \begin{cases} ab, & \text{ if } \ell\left(ab\right) = \ell\left(a\right) + \ell\left(b\right); \\ 0, & \text{ otherwise} \end{cases}$,
where $\ell\left(w\right)$ 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.)
