University of Birmingham > Talks@bham > Lab Lunch >  Characterizing po-monoids $S$ by completeness and injectivity of $S$-posets

Characterizing po-monoids $S$ by completeness and injectivity of $S$-posets

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  • UserMojgan Mahmoudi (Department of Mathematics, Shahid Beheshti University, Iran.)
  • ClockFriday 20 December 2013, 13:00-14:00
  • HouseCS 217.

If you have a question about this talk, please contact Neel Krishnaswami.

A poset with an action of a partially ordered monoid (po-monoid) $S$ on it, is called an $S$-poset, and $S$-poset maps are order-preserving maps which also preserve the action. Generalizing the notion of completeness of posets for $S$-posets we find two different notions of completeness for $S$-posets, one just completeness as a poset and the other poset completeness in such a way that the action distribute over arbitrary supremums. Then comparing these two notions, we find characterizations for some kinds of po-monoids.

Further, we study the notion of injectivity for $S$-posets. It is known that there are no nontrivial injective posets with respect to one-one order-preserving maps, we see that the same is true for $S$-posets. Although, Banaschewski and Bruns have shown that injective posets with respect to order-embeddings are exactly complete posets (Sikorski showed the same result for injective Boolean algebras). Therefore, we have also considered injective $S$-posets with respect to order-embedding $S$-poset maps, and proved that there are enough such injective $S$-posets. In fact, they are exactly the retracts of cofree $S$-posets over complete posets. But, we see that the counterpart of Banaschewski-Bruns theorem for posets is not necessarily true for $S$-posets. More precisely, injectivity of $S$-posets with respect to order-embedding $S$-poset maps implies the two above kinds of completeness, but the converse is not necessarily true. Then comparing these two notions for $S$-posets, again some homological classification of po-monoids and po-groups are obtained.

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