Also you are avoiding the actual "social" side of the Socialism. In Socialism, there is always a forced redistribution mechanism in the name of "exploitation of the labor class". Which makes the "free market" impossible.
The state can naturally also create monopolies. patents are explicit in this, granting an entity a "limited" timed monopoly.
> there is always a forced redistribution mechanism
And now we enter the definition of terms. Forced redistribution is mechanism, not a goal in itself. Historically there have been governments that have tried it, through to be honest there hasn't been a country that have not had events of forced redistribution, and if the book Debt - The First 5,000 Years is correct in the historical data, the elimination of debt through violence was a recurring event that happened in all cultures. Forced redistribution happens all the time in history when the ability to repay get too impossible low compared to the amount of money being owned.
Perhaps my wording was not correct. What I mean by "forced redistribution" is getting money from producers by force (via taxation) and distributing it to a arbitrarily selected people. This is required in all forms of Socialism and it violates the rules of free market (capitalism).
If we look far enough into the past, large enough monopolies are indistinguishable from governments. The Hanseatic League had their own armies, their own laws and even their own cities. Under their rule they fully controlled supply and demand, including price, which no longer makes it a free-market system. Still, they aren't what we define as a state (definitions...).
Taxation don't violate a free market. A free market need laws, which in turn require police and military, which in turn need to be paid (which distributes money). If everyone has to have their own police, military and laws, everyone becomes small islands of states. Those states then violate the supply-demand market, and thus you no longer have a free market.
There is no free lunch here. One can try to redefine what a free market is, what socialism is, what a state is, what a authority is, what taxation is, what interferences is, but it will just ends up arguing definitions.