amazon.de/…

I like this book, because it managed to teach me advanced material, which I already tried to learn with less success before.

Maybe I will buy it one day... But at least I have it in front of me now (the last time feels an eternity away.) I read chapter 7 Automatentheorie completely, the last time I had this book (if I remember correctly).

Has somebody here seen that that chapter 7 Automatentheorie, and knows comparable English texts?

I'd like to cite the definitions of deterministic and non-deterministic $M$-automata, where $M$ is an arbitrary monoid, such that the definitions are equivalent to the normal definitions if the monoid $M$ is free.

@babou Yes, that was my meaning. I gave the difference I see above; there is the field, and there are the objects. The discussion you propose has been happening for years, without definitive result.

@ThomasKlimpel I don't know the book, but I guess I could get it from the library. But then, I know no English texts about these basics. :/

Well, however basic M-automata are. (Is the algebraic approach even practiced internationally?)

Did you check articles of the authors? I'd assume they use the concept in their research as well.

(At least) One guy in my department works on similar topics; maybe this paper of his contains a useful reference?

His paper is written in English, so perhaps I'm lucky and can directly use his definitions.

Apart from that, the only "general purpose" reference from that paper seem to be the first edition from 1979 of "An Introduction to Automata Theory, Languages, and Computation" by John E. Hopcroft and Jeffrey D. Ullman. It seems to be completely different from later editions, and hard to obtain nowadays.

@ThomasKlimpel I think we have an old edition of that one flying around the group somewhere, but I don' think it covers monoids. I can have a look if you want, and if I find it.

@thomas hi. that is "algebraic theory of machines". it is a somewhat odd-fitting element/ area of CS/ math intersection that seems not largely studied anymore. the definitive/ comprehensive ref on this is Eilenberg from decades ago. he blazed a very unique trail few have followed.

amazon.com/Automata-languages-machines-Applied-Mathematics/dp/…

you might be able to follow other parts of this complex bio of eilenberg by may which cites the automata work almost as an afterthought/ footnote :|

An Appreciation of the Work of Samuel Eilenberg / May

(have found a scattered few other papers in this area & might try to dig em up if you relate to this stuff.)

elsewhere, this paper seems to connect with some directions J was pursuing at one pt

Shortest-Path Queries in Planar Graphs on GPU-Accelerated Architectures / Guillaume Chapuis, Hristo Djidjev

@vnz Your suggestion seems to fit. S. Eilenberg's Automata... is reference [32] in my German book, while I can't find Hopcroft et al in its reference section. The table of content of S. Eilenberg also looks very promising, the word monoid appears more than just once.

However, many of the references for chapter 7 seem to from 1965, and the Krohn Rhodes theorem from 1965 also explicitly mentions finite semigroups and machines, so I don't think that Eilenberg played any special role in the "algebraic theory of machines", appart from writting a widely acclaimed comprehensive reference of that field: