2

Let our signature be a single binary operation $\{*\}$. Consider the set of equational theories of that signature, partially ordered by inclusion. Is that partial order dense?

user107952
  • 20,508
  • 2
    I suspect you want your theories to be deductively closed, right (e.g. no distinction between ${xx=x}$ and ${yy=y}$)? So you're equivalently asking about the (opposite of the) partial order of varieties of magmas under inclusion. – Noah Schweber Jun 17 '22 at 23:00
  • 3
    The set of equational theories in a fixed language under the inclusion order forms an algebraic lattice. Any algebraic lattice of more than one element will have elements $a<b$ with no element strictly between $a$ and $b$. (Choose $b$ compact and $a$ a lower cover of $b$.) – Keith Kearnes Jun 17 '22 at 23:07
  • 2
    @KeithKearnes I think that should be an answer. – Noah Schweber Jun 17 '22 at 23:34
  • @NoahSchweber Yeah, by equational theory I mean deductively closed. – user107952 Jun 18 '22 at 01:12

1 Answers1

6

The set of equational theories in a fixed language under the inclusion order forms an algebraic lattice. Any algebraic lattice of more than one element will have elements $<$ with no element strictly between $$ and $$. (Choose $$ compact and $$ a lower cover of $$.)

Keith Kearnes
  • 13,798