Algebraic semantics is one type of semantics that uses algebraic expressions for connecting the formal descriptions of initial and final states of some operation of some operation that is defined in the programming language. I can not imagine more general and more versatile semantics, but yet - it is not used in today research. Why is that?
Traditionally symbolic methods were considered quite expensive, but current mathematical software, proof assistants and other automation of the mathematics and algebraic manipulations - all that provides the automation of the algebraic semantics. But still - algebraic semantics has dropped out of favour.
Why would we need algebraic semantics or semantics of programming languages generally? Answer is simple. This year the program synthesis field started to take off in earnest https://arxiv.org/search/?query=%22program+synthesis%22&searchtype=all&source=header . Apparently - if the search for target program could be done in the semantic domain (e.g. using relational/symbolic reinforcement learning or using inductive logic programming or other neuro-symbolic technique) and if there was the translation procedure from the formal semantics to whatever programming language, then the problem of program synthesis would be solved.
