Workers do not compete for jobs!

Arise, ye prisoners of starvation! Giles Merritt, editor of Europe's World and head of the Brussels-based think tanks "Friends of Europe" and "Security & Defense Agenda" has just announced that the suggestion workers compete for jobs is nonsense. I kid you not. Nor am I exaggerating. Mr. Merritt himself performed the reductio to his own absurdum at the Project Syndicate site:

Politicians and journalists often suggest that people compete for jobs, the implication being that bringing more women into Europe’s workforce would deny jobs to men. That is not true, of course: economists deride the idea that there is a given number of jobs to be divided up as the “lump of labor fallacy.” We need as many people working as possible, generating economic activity and paying taxes.

A careful reading of the above paragraph would show that the error resides in the transition -- or lack of one -- between the suggestion that people compete for jobs and the alleged implication that more women in the workforce denies jobs to men. Mr. Merritt has pulled that implication (along with the projected "idea that there is a given number of jobs to be divided up") out of his... "hat" This is the point about the so-called "lump-of-labor fallacy" that I have been trying to draw attention to for a decade and a half.

What are the implications for standard economic theory of the idea that workers do not compete for jobs?