On subamalgams of partially ordered monoids
Abstract: Previously one of us introduced a family of groups G M L (S), parametrized by a finite flag complex L, a regular covering M of L, and a set S of integers. We give conjectural descriptions of when G M L (S) is either residually finite or virtually torsion-free. In the case that M is a finite cover and S is periodic, there is an extension with kernel G M L (S) and infinite cyclic quotient that is a CAT(0) cubical group. We conjecture that this group is virtually special. We relate these three conjectures to each… Show more
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The main objective of this article is to study the ordered partial transformations PO ( X ) {\mathcal{PO}}\left(X) of a poset X X . The findings show that the set of all partial transformations of a poset with a pointwise order is not necessarily a pomonoid. Some conditions are implemented to guarantee that PO ( X ) {\mathcal{PO}}\left(X) is a pomonoid and this pomonoid is denoted by PO ↑ ( X ) {{\mathcal{PO}}}^{\uparrow }\left(X) . Moreover, we determine the necessary conditions in order that the partial order-embedding transformations define the ordered version of the symmetric inverse monoid. The findings show that this set is an inverse pomonoid and we will denote it by ℐPO ↑ ( X ) {{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X) . In case the order on the poset X X is total, we explore some properties of PO ↑ ( X ) {{\mathcal{PO}}}^{\uparrow }\left(X) and ℐPO ↑ ( X ) {{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X) , including regressive, unitary, and reversible.
The main objective of this article is to study the ordered partial transformations PO ( X ) {\mathcal{PO}}\left(X) of a poset X X . The findings show that the set of all partial transformations of a poset with a pointwise order is not necessarily a pomonoid. Some conditions are implemented to guarantee that PO ( X ) {\mathcal{PO}}\left(X) is a pomonoid and this pomonoid is denoted by PO ↑ ( X ) {{\mathcal{PO}}}^{\uparrow }\left(X) . Moreover, we determine the necessary conditions in order that the partial order-embedding transformations define the ordered version of the symmetric inverse monoid. The findings show that this set is an inverse pomonoid and we will denote it by ℐPO ↑ ( X ) {{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X) . In case the order on the poset X X is total, we explore some properties of PO ↑ ( X ) {{\mathcal{PO}}}^{\uparrow }\left(X) and ℐPO ↑ ( X ) {{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X) , including regressive, unitary, and reversible.
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