Dice Question Streamline Icon: https://streamlinehq.com

Prime objects among finite algebras of type <1,1>

Determine whether there exists any prime object, in Tarski’s sense, in the class of all finite algebras of type <1,1> (algebras with two unary operations), or in any algebraic type that has more than one operation of arity at least 1.

Information Square Streamline Icon: https://streamlinehq.com

Background

The paper cites two questions from [6] that remain open; the second asks about prime objects in finite algebras of type <1,1>, which can be viewed as sets with an action of the free semigroup on two generators.

The authors explicitly confirm these questions remain open and present the query verbatim.

References

So far as I know, the other two questions remain open. Question 5.2. [6, p.263] Is there any prime in the class of all finite algebras of type <1,1> (or in any type that has more than one operation of arity ≥ 1)?

— On semigroups that are prime in the sense of Tarski, and groups prime in the senses of Tarski and of Rhodes (2409.15541 - Bergman, 23 Sep 2024) in Question 5.2, Section 5