Is there a word analogous to "dual" for three or more options?

You can use "threefold", "fourfold", and if allowing for even more options or composed by any more parts, I'd suggest "multiple" or "manifold".

There seems to be no way of directly continuing the etymological pattern that formed 'dual' in English. Dual is derived from the latin dualis.

The equivalent number for for three in latin would be tres or tria. In English, we have the word 'trial'. We also have the prefix tri-, but that is not a word in and of itself.

In suspecting that trial would probably be the closest analog, I did come across 'paucal', which is listed as meaning:

paucal

Adjective

…

2. (grammar) pertaining to a language form referring to a few of something (three to around ten), as a small group of people; singular, dual, trial and plural.

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Noun

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3.(uncountable)(grammar) a language form referring to a few of something (three to around ten), as a small group of people; contrast singular, dual, trial and plural.

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Excerpts from the Wikitionary entry of Paucal, licensed under the CC-SA-3.0 License.

It should be noted that I am not recommending 'paucal' as an alternative. It is a technical word with a restricted purpose that does not really match the desired definition, especially since it is less specific.

The reason I included the Wiktionary definitions of 'paucal', despite not recommending it, is that it suggests 'trial' would be the word to continue the pattern but again, if it were not for a couple of problems:

The linguistic word I found on Wiktionary suggests that a similar definition exists for trial, but that definition does not match either. I think this is actually more clearly demonstrated by the closest definition in the Randomhouse Kernerman Webster's College Dictionary, © 2010 definition of trial:

adj. 1. of or belonging to a grammatical category of number, as in some Papuan and Austronesian languages, used to indicate that a word denotes three persons or things.

More importantly, as I am sure you are all already well aware, 'trial' usually means something closer to "test" in English, and perhaps most especially the sort of test that requires adjudication in a court of law. I expect the word to be such a popularly and traditionally pervasive word that it is hard for me to endorse using the word to mean anything else, except perhaps by way of metaphor or metonymy.

As far as I can reliably discern, 'trial' does not even derive from tria, but the anglo-french triet which means something closer to try, as in an effort or attempt. Thus if you want a word which is in the same series as 'dual' that conveys the meaning you desire, I am currently of the opinion that a such word does not exist.

You may want to consider 'plural' as a possible alternative, but it is not as specific:

plural, adjective [Latin pluralis, from plus, pluris, more.]

1. Containing more than one; consisting of two or more, or designating two or more; as a plural word.

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The American Dictionary of the English Language by Noah Webster

Assuming you only had three roles and no more, perhaps the adjective triadic from triad would work.

The circulatory system plays a three-way role of carrying oxygen to the body's tissues...

This would be the most well-known way to phrase to sentence, it's very colloquial and not so scientific but nearly everyone will understand what you mean.

The word triune means three-in-one. It is most notably used in reference to the Holy Trinity, as

the triune Deity

referring to the Father, the Son, and the Holy Spirit.

However, it's use can be generalized to mean the same thing as dual, but for three.

I was looking for the same thing, specifically for 3, and came across trinal. (For example, the aeroplane had dual controls.)

Re above suggestions: triple and three-fold have the connotation of identical instances or a multiplication of a single instance, which dual does not. (They also have the limitation of being related to double and twofold.) Three-way has connotations of some sort of physical object or direction, that I don't want either. Triadic might work but it is of Greek origin, whereas dual and trinal are from Latin.

Paucal, is not specific to three, but is a good alternative to multiple.

The trouble with multiple or multiplex is that they don't exclude duality.

A word that does exclude two-ness from a sense of (moderate) multiplicity is several or severalty.

However, you couldn't say something performed "a several role", you'd have to say "several roles". Or, at the risk of contrivance, "a severalty of roles..."

trichotomic. : belonging to, characterized by, or based upon a trichotomy

trichotomous : divided or dividing into three parts or into threes

tri·chot·o·my : a division into three categories.

Sources respectively: Merriam Webster, Merriam Webster, Google