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 f-algebra [2014/09/23 10:34]nikolaj f-algebra [2014/09/23 10:36] (current)nikolaj Both sides previous revision Previous revision 2014/09/23 10:36 nikolaj 2014/09/23 10:34 nikolaj 2014/09/23 10:32 nikolaj 2014/09/22 23:21 nikolaj 2014/09/22 23:19 nikolaj 2014/09/22 23:15 nikolaj 2014/09/22 23:15 nikolaj 2014/09/22 23:14 nikolaj old revision restored (2014/09/22 23:00)2014/09/22 23:14 nikolaj 2014/09/22 23:00 nikolaj 2014/09/22 22:59 nikolaj 2014/09/22 22:17 nikolaj 2014/09/22 19:05 nikolaj old revision restored (2014/09/22 18:37) 2014/09/23 10:36 nikolaj 2014/09/23 10:34 nikolaj 2014/09/23 10:32 nikolaj 2014/09/22 23:21 nikolaj 2014/09/22 23:19 nikolaj 2014/09/22 23:15 nikolaj 2014/09/22 23:15 nikolaj 2014/09/22 23:14 nikolaj old revision restored (2014/09/22 23:00)2014/09/22 23:14 nikolaj 2014/09/22 23:00 nikolaj 2014/09/22 22:59 nikolaj 2014/09/22 22:17 nikolaj 2014/09/22 19:05 nikolaj old revision restored (2014/09/22 18:37) Line 17: Line 17: * Addition of natural numbers is a binary relation: $+:​\mathbb{N}\times\mathbb{N}\to\mathbb{N}$. Hence $\langle \mathbb{N},​+\rangle$ is an $F$-algebra for the endofunctor with object map $FX:​=X\times X$. * Addition of natural numbers is a binary relation: $+:​\mathbb{N}\times\mathbb{N}\to\mathbb{N}$. Hence $\langle \mathbb{N},​+\rangle$ is an $F$-algebra for the endofunctor with object map $FX:​=X\times X$. - * Group actions ​on $X$ are maps $m:G\times X\to X$, so consider $F_GX:=G\times X$. The first example is also an $F_\mathbb{N}$-algebra. + * Fix a monoid $M$. A monoid action ​on $X$ is a map $\alpha:M\times X\to X$, so consider $FX:=M\times X$. Incidentally, ​$\langle ​\mathbb{N},+\rangle$ can also be viewed as an $F$-algebra ​for $M=\mathbb{N}$. === Reference === === Reference ===