The very insightful questions that arise, when someone studies micro foundation are the following: What is the difference between competitive and complete marekts? When we refer to the first is it equivalent as to refer in the latter? Also, when we speak about non-competitive markets, can we also refer to them as incomplete markets? In addition to this, complete or incomplete markets have always the so-called Arrow-Debreu sense of complete/incomplete markets?
Complete market is a market where every possible asset or good can be assigned a price and where you have perfect information, can make perfect contracts and zero transaction costs. Any market can be complete regardless of its market structure. So you can have complete market dominated by monopoly, or oligopoly or monopolistic competition etc.
Perfectly competitive market is a market where there are so many buyers and seller that firms become price takers and when the products are identical so there is a little scope for differentiation.
You can’t just call non-competitive market incomplete. For example, in standard undergrad treatment of monopoly you still assume complete markets, yet the market structure is non-competitive.
Yes when people talk about complete markets they do it in a pretty much same sense as Arrow-Debreu