Suppose we have a monoid M with a "star" operation * : M -> M satisfying a* = aa*. I'm not sure if there are other laws we might also want (e.g. perhaps we also want a* = a*a), but the idea is that a* is supposed to represent "combining a with itself infinitely often", i.e. the limit \(\lim_{n \to \infty} a^n\). One simple example would be the natural numbers adjoined with positive infinity under addition, where 0* = 0 and n* = infinity for every n > 0. Another example is the booleans under conjunction or disjunction, where false* = false and true* = true. The booleans under xor, on the other hand, do not have a star operation.
Does this have a name? I found the concept of a "complete monoid", but that seems stronger than what I want: a complete monoid allows making sense of arbitrary infinitary products, but I only want to be able to take infinite powers (though off the top of my head I can't think of an example monoid that has a star but is not complete...)
Brent Yorgey
npub1g8…a88a8
2026-06-24 22:28:50 UTC
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2026-06-24 22:28:50 UTCEvent JSON
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"content": "Suppose we have a monoid M with a \"star\" operation * : M -\u003e M satisfying a* = aa*. I'm not sure if there are other laws we might also want (e.g. perhaps we also want a* = a*a), but the idea is that a* is supposed to represent \"combining a with itself infinitely often\", i.e. the limit \\(\\lim_{n \\to \\infty} a^n\\). One simple example would be the natural numbers adjoined with positive infinity under addition, where 0* = 0 and n* = infinity for every n \u003e 0. Another example is the booleans under conjunction or disjunction, where false* = false and true* = true. The booleans under xor, on the other hand, do not have a star operation.\n\nDoes this have a name? I found the concept of a \"complete monoid\", but that seems stronger than what I want: a complete monoid allows making sense of arbitrary infinitary products, but I only want to be able to take infinite powers (though off the top of my head I can't think of an example monoid that has a star but is not complete...)",
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