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Turkish Journal of Mathematics

DOI

10.55730/1300-0098.3468

Abstract

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 semigroups has as intersection the ordered semigroup $\left( U,{{\cdot }_{U}},{{\le }_{U}} \right)$ which is a subsemigroup of $\left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right)$ for every $i\in I$ (with respect to the operation and order relation of each $\left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right)$, $i\in I$). To do this, we first consider the ordered semigroup amalgam $\mathfrak{A}=\left[ \left\{ \left( {{S}_{i}},{{\cdot }_{i}},{{\le }_{i}} \right),i\in I \right\};\left( U,{{\cdot }_{U}},{{\le }_{U}} \right);\left\{ {{\varphi }_{i}}:U\to {{S}_{i}},i\in I \right\} \right]$ (where $\left\{ {{\varphi }_{i}}:U\to {{S}_{i}},i\in I \right\}$ is a family of monomorphisms) and then we construct the free ordered product $\underset{i\in I\text{ }}{\mathop{\Pi _{U}^{*}}}\,{{S}_{i}}$ of the ordered semigroup amalgam $\mathfrak{A}$ considering the ordered quotient of the free ordered product ${{\underset{i\in I}{\mathop{\Pi }}\,}^{*}}{{S}_{i}}$ by an appropriate pseudoorder of ${{\underset{i\in I}{\mathop{\Pi }}\,}^{*}}{{S}_{i}}$ through which for each $i,j\in I$ and for each $u\in U$, ${{\varphi }_{i}}\left( u \right)\in {{S}_{i}}$ is identified (by means of monomorphisms) with ${{\varphi }_{j}}\left( u \right)\in {{S}_{j}}$. We give a sufficient and necessary condition so that an ordered semigroup amalgam is embedded in an ordered semigroup. At the end of the paper, we introduce the notion of ordered dominions. An element $d$ of an ordered semigroup $S$ is dominated by a subsemigroup $U$ of $S$ if for all ordered semigroups $\left( T ,\cdot ,\le \right)$ and for all homomorphisms $\beta ,\gamma :S\to T$ such that $\beta \left( u \right)=\gamma \left( u \right)$ for each $u\in U$, we have $\left[ \beta \left( d \right) \right)_{\le }^{T}\cap \left[ \gamma \left( d \right) \right)_{\le }^{T}\ne \varnothing $. In the last Theorem of the paper, we give an expression of the set of elements of $S$ dominated by $U$ based on ordered semigroup amalgams.

Keywords

Ordered semigroup, pseudoorder on an ordered semigroup, ordered quotient of an ordered semigroup by a pseudoorder, free ordered product of ordered semigroups, ordered semigroup amalgam, free ordered product of an ordered semigroup amalgam, ordered dominions

First Page

1857

Last Page

1872

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