Abstract:
Motivated by the recent definition of the $AM$-property
in locally solid vector lattices [O. Zabeti, doi:
10.1007/s41980-020-00458-7], in this note, we try to investigate
some counterparts of those results in the category of all locally
solid lattice rings. In fact, we characterize locally solid lattice
rings in which order bounded sets and bounded sets agree.
Furthermore, with the aid of the $AM$-property, we find conditions
under which order bounded group homomorphisms and different types of
bounded group homomorphisms coincide. Moreover, we show that each
class of bounded order bounded group homomorphisms on a locally
solid lattice ring $X$ has the Lebesgue or the Levi property if and
only if so is $X$.
Key words:locally solid lattice ring,
bounded group homomorphism, $AM$-property, Levi property, Lebesgue property.
Fulltext:
PDF file (236 kB)