Consider $S$ a bounded semilattice (a partially ordered set $S$ with binary joins $s\lor t$ and a top element $\top \geq s$).
Definition: Call $t\in S$ transitive when for every $s,s'\in S$, if $s\lor t\lneq\top$ and $t\lor s'\lneq\top$ then $s\lor s'\lneq\top$. Dually: $t$ is transitive when $s{\lor} s'=\top$ implies $(s{\lor} t=\top) \lor (t{\lor} s'=\top)$.
Question: Does anybody recognise this property, or can anyone relate it to known and studied properties in mathematics or logic?
Thanks.
Let $[t,\top]=\{x\in S\;|\;t\leq x\leq \top\}$ be the interval in $S$ between $t$ and $\top$. It is not hard to show that the transitivity of $t$ in $S$ is equivalent to the property that the top element of the interval $[t,\top]$ is join-irreducible.
Here are two places in mathematics where terminology has been coined to discuss intervals with join-irreducible top element.
Equational logic. Let $L$ be an algebraic first-order language and let $S$ be the lattice of equational theories in the language $L$. ($S$ is ordered by inclusion.) The top element $\top$ of $S$ is the theory consisting of all consequences of $x\approx y$, which is the theory of all equations in the language $L$. An element $t\in S$ is called precomplete if $\top$ is a join-irreducible element of $[t,\top]$. Hence, the word precomplete has been used to describe what is called transitive in the question above.
This example is a little bit misleading, because the compactness theorem implies that if $[0,\top]$ has a join-irreducible top element, then it has a completely join-irreducible top element. Thus a precomplete theory is a consistent theory with a unique completion. This fact makes the word precomplete seem like a natural choice of terminology in this setting. The definition of transitivity from the question does not involve any completeness assumption. Thus, precomplete might not be a satisfactory substitute for transitive in all cases. (It will be a satisfactory substitute in all cases where $S$ is a finite join semilattice. For finite $S$, an element $t\in S$ is transitive exactly when $t$ is dominated by a unique coatom of $S$.)
Module theory. A module $M$ is called hollow if every proper submodule is small. This is equivalent to the property that the top element of the submodule lattice $\textrm{Sub}(M)$ is join-irreducible. In the language of the question, hollowness expresses that $0$ is a transitive submodule of $M$. (In fact, it implies that every submodule of $M$ is a transitive element of $\textrm{Sub}(M)$.)
More generally, a submodule $N\leq M$ is called a hollow submodule of $M$ if the top element of the lower interval $[0,N]$ in $\textrm{Sub}(M)$ is join-irreducible. The transitive property concerns upper intervals instead of lower intervals, and I don't know a common term for the situation when $[N,M]$ has join-irreducible top element. Of course, $N$ has this property iff $M/N$ is hollow. Perhaps a ring theorist might say that $N$ is co-hollow or dually hollow.