02/15/2017, 10:07 PM
ok, take your time.
Anyways I think this is only the tip of the iceberg.
About clone theory ... this is insanely curious. This is not a theory that you meet that often... at least I don't. This is the second time I hear of that and the first time was... inside an answer an MSE user gave me about compatible posets on algebraic structures.. to be more precise I had in mind the transitive closure of dynamical systems (defined by left or right translations of binary operations) ... and he came up with the fact that it is sufficient to look at the compatibility of quasiorders on monounary algebras... that are just dynamical systems where the phase space need not to be a metric space or a topological one.
Obviously I went into this because monounary algebras are the synthetic way to talk about ranks in full abstraction. There you can define connectedness, define a meet operation, talk about three kind of ranks too (finite, period, "co-rank" and rank relative to an element) and alot of other interesting notions form a purely algebraic point of view that is more fundamental than the topological features of dynamical systems. (fixed points, cyclic/periodical points, the lattice of the iterations are all algebraic in nature. look)
And Clone theory seems so unrelated to all of this that I do not know well what to think.
Anyways I think this is only the tip of the iceberg.
About clone theory ... this is insanely curious. This is not a theory that you meet that often... at least I don't. This is the second time I hear of that and the first time was... inside an answer an MSE user gave me about compatible posets on algebraic structures.. to be more precise I had in mind the transitive closure of dynamical systems (defined by left or right translations of binary operations) ... and he came up with the fact that it is sufficient to look at the compatibility of quasiorders on monounary algebras... that are just dynamical systems where the phase space need not to be a metric space or a topological one.
Obviously I went into this because monounary algebras are the synthetic way to talk about ranks in full abstraction. There you can define connectedness, define a meet operation, talk about three kind of ranks too (finite, period, "co-rank" and rank relative to an element) and alot of other interesting notions form a purely algebraic point of view that is more fundamental than the topological features of dynamical systems. (fixed points, cyclic/periodical points, the lattice of the iterations are all algebraic in nature. look)
And Clone theory seems so unrelated to all of this that I do not know well what to think.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
