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Full-Text Articles in Physical Sciences and Mathematics
Free Ordered Products-Ordered Semigroup Amalgams-Ordered Dominions, Michael Tsingelis
Free Ordered Products-Ordered Semigroup Amalgams-Ordered Dominions, Michael Tsingelis
Turkish Journal of Mathematics
Given an indexed family $\left\{ \left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right),i\in I \right\}$ of disjoint ordered semigroups, we construct an ordered semigroup having $\left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right)$, $i\in I$ as subsemigroups (with respect to the operation and order relation of each $\left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right)$, $i\in I$). This ordered semigroup is the free ordered product ${{\underset{i\in I}{\mathop{\Pi }}\,}^{*}}{{S}_{i}}$ of the family $\left\{ {{S}_{i}},i\in I \right\}$ and we give the crucial property which essentially characterizes the free products. Next we study the same problem in the case that the family $\left\{ \left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right),i\in I \right\}$ of ordered …
What Can Lattices Do For Hypersemigroups?, Niovi Kehayopulu
What Can Lattices Do For Hypersemigroups?, Niovi Kehayopulu
Turkish Journal of Mathematics
This is from Birkhoff, the "father of lattice theory" in Trends in Lattice Theory. Van Nostrand 1970: "Lattices can do things for you, no matter what kind of mathematician you are!". The aim of this paper is to show that the $le$-semigroups (lattice ordered semigroups possessing a greatest element) play the main role in studying the ordered hypersemigroups. From many results on lattice ordered semigroups corresponding results on ordered semigroups can be obtained. The converse is also possible but the beauty and simplicity of "order" makes it easier to investigate the lattice ordered semigroup at first. After getting the results …