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Let $(G,\cdot)$ be a partial magma (a set endowed with a partial binary operation). In principle, for such generic structures it is possible that $\exists g \in G$ such that $\forall h \in G, \, g\cdot h \not \in G$.

On the other hand, closure is a requirement such that $\forall g,h \in G, \, g\cdot h \in G$.

I am faced with an algebraic structure for which $\forall g \in G, \exists h \in G$ such that $g\cdot h \in G$. In other (colloquial) words, for any element there is always something to multiply that element with that will fall in $G$.

This is somewhat halfway between the two extremes - is there a name for such a property? Something like weak closure?

I am thinking that this is probably something useful for partial semigroups as well - where associativity is relaxed to those elements which are closed under multiplication.

Thank you!

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    $\begingroup$ The "multipliable" relation is (left-)total? $\endgroup$
    – Deusovi
    Commented Jun 13 at 19:45
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    $\begingroup$ I think writing "$g\cdot h\notin G$" is misleading. The issue is that $g\cdot h$ may not be defined, not that it is not an element of $G$. There is a difference between "closure" (which is subsumed by calling $\cdot$ a partial operation) and whether the partial operation is defined or not. We usually say that $\frac{1}{0}$ is "not defined", not that "it is not an element of $\mathbb{R}$". $\endgroup$
    – Arturo Magidin
    Commented Jun 14 at 5:29
  • $\begingroup$ So it's not "closed" or "closure". I think you are misusing the term throughout. $\endgroup$
    – Arturo Magidin
    Commented Jun 14 at 5:31

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