COFINAL SIMPLICITY AND ALGEBRAIC
Paul D. Bacsich · 1972
We shall find a syntactic characterisation of simplicity and use it to prove the following result: if T is an equational theory with enough simple algebras and n is a cardinal, then every n-algebraically closed T-algebra is n-existentially closed and simple. By similar methods we then prove that in any countable equational T the class of absolute retracts is of countable character. Finally by applying methods analogous to Higman-Neumann-Neumann extensions we show that every algebraically closed lattice is simple.