Y by ishan.panpaliya, felixgotti
Problem Statement: Prove that a monoid is a GL-monoid if and only if it is prime-like.
Proof:
Prime-like GL:
Let be a prime-like monoid. Suppose such that . Suppose further towards a contradiction that ; that is, that there exists some divisor with . Then because is prime-like, has some divisor with either or . Either way, we also must have , which means that either or . This is a contradiction. Thus, , which means is GL.
GL prime-like:
Let be a GL monoid, and suppose with . Assume towards a contradiction that for all , and (in other words, shares no divisors with or ). This implies that . Since is GL, this means that . This contradicts the assumption that . Thus, there must exist some with or , which means is prime-like.
Proof:
Prime-like GL:
Let be a prime-like monoid. Suppose such that . Suppose further towards a contradiction that ; that is, that there exists some divisor with . Then because is prime-like, has some divisor with either or . Either way, we also must have , which means that either or . This is a contradiction. Thus, , which means is GL.
GL prime-like:
Let be a GL monoid, and suppose with . Assume towards a contradiction that for all , and (in other words, shares no divisors with or ). This implies that . Since is GL, this means that . This contradicts the assumption that . Thus, there must exist some with or , which means is prime-like.