Is there any research on Turing machines with transition relation homomorphic to given algebraic structure? A Turing machine is defined as a structure $ TM(L,Q,T) $, where $L,Q$ are sets of symbols and internal states of TM respectively, and T is a transition relation:

$T: L \times Q \to L \times Q $ 

for simplicity suppose that $Q$ contains all "terminal states" that is $ \{ START,STOP \} $, and information about next move of the head on the tape, that is $\{L,R\}$ - it is just another information about internal state of the machine. $L$ contain all terminal symbols. This are finite sets. 

This assumptions are for simplicity and easy notation, because full definition of TM is well known, and there is no need to explain it I presume. Please take a look for other questions below where such simplification is provided in more detailed way.

$T$ is then a table with for example elements $Q$ in the column and elements of $L$ in  the rows. An element of such table is value of $T$ for its arguments that is:

$T(q_i,a_k) = (q_j,a_s)$

Suppose that there is a homomorphism between $T(q_i,a_k)$ (as table) and certain finite algebraic structure (group, monoid, algebra, possibly some exotic structure). Or suppose that such transition relation has certain symmetries (for a stupid example here let me write: $T(q_i,a_k) = T(q_k,a_i)$) or "subtables" with structure or symmetries homomorphic to given algebraic or other structure.


For given structure of $T(q_i,a_k)$,
are there any known facts about
class    of TM with that structure?
Are there any known equivalence
classes of $T(q_i,a_k)$ arising from algebraic
properties of $T(q_i,a_k)$ ?




Asking this question I follow the suggestion of  Dave Clarke. This is question related to following questions: 


asked here - Turing Machine which generates order on the set of its states 
asked on mathoverflow (and for which there is an answer by Joel David Hamkins) Turing Machine which generates order on the set of its states
There are results about algebraic properties of languages (as opposed to the machines accepting the languages). For example, regular languages can be characterised using monoids. The Myhill-Nerode theorem relates an algebraic structure with the transition relation of a specific class of finite automata. You could look up those and subsequent results. 

Rather than homomorphisms, bisimulation and simulation relations are the notion of choice for relating the structure of transition systems. Checking language equivalence between pushdown automata is undecidable but deciding equivalence of languages accepted by deterministic PDAs was open until 2001 when Gerard Sénizergues showed it was decidable. His proof uses algebraic ideas. Colin Stirling later gave a simplified proof (writeups and slides are available on his homepage). 


  Sénizergues, Géraud, The equivalence problem for deterministic pushdown automata is decidable, Degano, Pierpaolo (ed.) et al., Automata, languages and programming. 24th international colloquium, ICALP ’97, Bologna, Italy, July 7--11, 1997. Proceedings. Berlin: Springer-Verlag (ISBN 978-3-540-63165-1/pbk; 978-3-540-69194-5/ebook). Lecture Notes in Computer Science 1256, 671-681 (1997). ZBL1401.68168.