Is there a concept representing "connectedness" in abstract algebra? Consider an object, call it a web, that consists of a set $S$ equipped with a binary operation obeying these axioms:
$$
\forall\ a,b \in S\ \exists\ c \in S :a\ \bullet\ b=c
$$
$$
\forall\ a,b \in S\ \exists\ c \in S : a\ \bullet\ c=b
$$
The first axiom is just closure, but the second axiom is not one that I'm aware of in any of the literature I've checked. I'll call it connectedness in lieu of any formal language I'm unaware of. It does obviously arise as a logical consequence of identity and invertibility but those are not necessary. Case in point, consider the set $S=\{A,B,C,D\}$ and a binary operation $\bullet$ obeying the following multiplication table:




$\bullet$
$\boldsymbol{A}$
$\boldsymbol{B}$
$\boldsymbol{C}$
$\boldsymbol{D}$




$\boldsymbol{A}$
$A$
$C$
$B$
$D$


$\boldsymbol{B}$
$C$
$A$
$D$
$B$


$\boldsymbol{C}$
$D$
$B$
$A$
$C$


$\boldsymbol{D}$
$B$
$D$
$C$
$A$




This set is obviously closed under this operation, however it has no identity element. This operation is also not associative:
$$
(C \bullet B) \bullet C = D \bullet C = C
$$
$$
C \bullet (B \bullet C) = C \bullet B = D
$$
This object is therefore not a monoid. It is a magma, because of the closure axiom all webs would be magmas, but the inverse isn't true. The set of all natural numbers is a magma under addition, but is not a web because $5+x=3$ has no solution in the natural numbers.
I would love to research this topic more but without knowing the correct terminology I haven't made much progress. Can anyone tell me if this concept is already known and what things to look for to find more information?
 A: One way to phrase your property is to define, if $S$ is your domain, functions $\phi_a:S\to S,$ with $\phi_a:x\mapsto a\bullet x.$ Then your property is that each $\phi_a$ is onto.
The property that each $\phi_a$ is one-to-one has a name, left cancellative.
But your sort-of dual property does not seem to have a name, at least not one listed on Wikipedia’s list of binary operation properties.
Cancellative might have a name because it applies to the foundational $\mathbb N^{>0}$ under both $+$ and $\times.$

One reason I dislike “connected” is that it feels related to graph theory, but a graph is connected if there is a sequence of edges. So in a graph theory sense, $a$ would be (left)-connected to $b$ if there was a $c_1,c_2,\dots,c_n$ such that $a\bullet c_1\bullet\cdots\bullet c_n=b,$ where the operations are performed left to right.
I suggested in comments the term “right transitive,” in analogy to group actions. But that terminology might be almost too strongly tied to groups, giving an implication of associativity and inverses.
Another possible term, if we are thinking about graph theory, is “left-complete.” A directed graph is complete if all ordered pairs of nodes have an edge between the first and second node.

On a finite set $S,$ a function $f:S\to S$ is one-to-one if and only if it is onto. So for finite sets, left-cancellative is equivalent to left-complete. Together, they imply left-divisible. Same for the right- versions, of course. So, on a finite set, a (both left-and-right) complete operation implies that $(S,\bullet)$ is a quasigroup.
