I think some MUDs could qualify better than others. A key element (IMO) of a Roguelike is randomness in the maps/scenarios, not just the encounters. I say "key" in the sense of "necessary but not sufficient". Most MUDs have fixed maps, perhaps with a few small areas that are randomized. So you can memorize the layouts of many, if not all, areas. There is exploration, and you may forget areas, but the exploration is really a one time thing. In Nethack, by contrast, you really don't know what's coming next (though sometimes it reloads a level you died on before [0]).

[0] I was killed by a mummy. And in my next run found my now mummified former self and the mummy. After a few runs that level was full of mummies. Needless to say, it did not go well for me.

Is the map being random an important quality? What if the map was fixed but all the enemies and items shuffled about... If that doesn't count are maps with rooms (i.e. hallways and wide open spaces) where enemies only spawn in the open space... if they can be pretty trivially reduced into a graph that another generation reduces into then is it the same map?

I do agree that random maps are sort of a hallmark of classic roguelikes to me, but roguelike is such a slippery thing to define.

It's an interesting question. Different things can be randomized and to different degrees and it kind of gives you different games. Suppose only RPG-like mechanics (PC has stats, they level up with encounters, etc.).

If you have a (roughly) fixed map where the PC-level appropriate mobs spawn randomly, so you always know the path to victory (or paths) but the challenge is subtly altered by these random encounters, I think you have a JRPG.

If you have a roughly fixed map, random mobs, but which path to take gets randomized (left leads to the boss in one run, right in another run) you're pressing into roguelike territory. But you could also call it Hunt the Wumpus: The RPG.

Roguelikes, again this is opinion, significantly randomize the map as well as having random encounters. I think it changes the way the game is played and explored and why it's replayed.

In the first, you replay the game because you want to try focusing on different stats or classes, an optimal build, the story was fun, etc.

In the second, you replay it because of the general challenge, maybe a bit about optimal builds and such, but also (to whatever extent there is) the puzzling nature of it. I hear a wind in this room, so that means the pit is in one of those two. Better be careful.

In the third, you replay it for all of those reasons and because you genuinely don't know what's coming next. You can have a game where there are a series of holes in the ground and your 2nd level character is suddenly 20 dungeon levels down. Well shit, can my tourist actually survive this?

And these are only three discrete points along a wide spectrum of randomness and its utilization in a game. I think roguelikes tend to be more roguelike when the randomness is somewhere above the second example, up to the extreme of the third. But you could also find other elements of the game to randomize that give you non-physical spaces to explore in a similar manner.

Again, randomness isn't sufficient, but it is necessary. And the degree of randomness or its utilization can be reduced if you keep other "roguelike" qualities present in the game.

Roguelikes tend to have a text-based map, like: you have a location on an X-Y grid; some cells of the grid have walls and things. Also, time moves forward at your own pace.

MUDs vary widely, but typically operate by discretizing things into "rooms", often with description in writing rather than as a map layout; the various "rooms" form a graph. If you've played Colossal Cave Adventure, like that (but with fewer puzzles and more combat). Also, time moves forward at the server's preferred pace.

To clarify I think you mean roguelikes have maps where the character is the size of a pixel of map information - so you can't stand in a position that only partially overlaps with the position you previously stood in... all game forever (probably?) will have coordinates the big difference is how substantial that coordinate system is compared to the size of the player... having pixel sized players makes a lot of stuff really easy like detecting drowning by seeing if the pixel you're in is flooded - but it's really unimportant on a gameplay level... it just simplifies the tech needed to build out the engine.

No, the thing I wanted to highlight was that often MUDs do not have coordinates. There are (player-entity -> room) associations instead: more of a graph than a representation of space.