Algebraic automata theory is very cool.

Books

• The Mathematical Foundations of Automata Theory

Papers

• Decomposition of Automata (MIT): people.csail.mit.edu/meyer/remarks-on-algebraic-decomposition.pdf
• Krohn Rhodes Logics: arxiv.org/pdf/2304.09639.pdf

Structures

In algebra, we kind of realize there are lots of different types of structures, and all of them enforce different types of rules. Here's a nice list of algebraic objects as a prelude to algebraic automata theory.

• Category: simply, a collection of objects that are linked together by arrows. You can compose the arrows associatively, and each object has an identity arrow. A basic example is the category of sets, where objects are sets, and arrows are functions.

Group-like (one binary operation):

• Magma: A set with a closed binary operation
• Semigroup: A set with a closed and associative binary operation
• Monoid: A set with an identity, and the binary operation is closed, associative, and invertible
• Group: A set with an identity, and the binary operation is closed, associative, and invertible
• Abelian Group: A set with an identity, and the binary operation is closed, associative, invertible, and commutative

Ring-like:

• Ring: A set with two binary operations (written addition and multiplication) such that commutativity, associativity, inverse, and identity exist for addition, and there's distribution over addition
• Semiring: A ring, but with no additive inverse property
• Field: A ring that additionally has multiplicative identity, multiplicative inverse, and multiplicative commutativity

For Algebraic Automata Theory, Semigroups and Identities are key!