Unbounded Order Convergence in Ordered Vector Spaces

We consider an ordered vector space <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>X</a:mi></a:math> . We define the net <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mfenced open="{" close="}" separators="|"><c:mrow><c:msub><c:mrow><c:mi>x</c:mi></c:mrow><c:mrow><c:mi>α</c:mi></c:mrow></c:msub></c:mrow></c:mfenced><c:mo>⊆</c:mo><c:mi>X</c:mi></c:math> to be unbounded order convergent to <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"><h:mi>x</h:mi></h:math> (denoted as <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"><j:msub><j:mrow><j:mi>x</j:mi></j:mrow><j:mrow><j:mi>α</j:mi></j:mrow></j:msub><j:mover accent="true"><j:mrow><j:mo>⟶</j:mo></j:mrow><j:mrow><j:mtext>uo</j:mtext></j:mrow></j:mover><j:mi>x</j:mi></j:math> ). This means that for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M5"><m:mn>0</m:mn><m:mo>≤</m:mo><m:mi>y</m:mi><m:mo>∈</m:mo><m:mi>X</m:mi></m:math> , there exists a net <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M6"><o:mfenced open="{" close="}" separators="|"><o:mrow><o:msub><o:mrow><o:mi>y</o:mi></o:mrow><o:mrow><o:mi>β</o:mi></o:mrow></o:msub></o:mrow></o:mfenced></o:math> (potentially over a different index set) such that <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M7"><t:msub><t:mrow><t:mi>y</t:mi></t:mrow><t:mrow><t:mi>β</t:mi></t:mrow></t:msub><t:mo>↓</t:mo><t:mn>0</t:mn></t:math> , and for every <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M8"><v:mi>β</v:mi></v:math> , there exists <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M9"><x:msub><x:mrow><x:mi>α</x:mi></x:mrow><x:mrow><x:mn>0</x:mn></x:mrow></x:msub></x:math> such that <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M10"><z:msup><z:mrow><z:mfenced open="{" close="}" separators="|"><z:mrow><z:mo>±</z:mo><z:msup><z:mrow><z:mfenced open="(" close=")" separators="|"><z:mrow><z:msub><z:mrow><z:mi>x</z:mi></z:mrow><z:mrow><z:mi>α</z:mi></z:mrow></z:msub><z:mo>−</z:mo><z:mi>x</z:mi></z:mrow></z:mfenced></z:mrow><z:mrow><z:mi>u</z:mi></z:mrow></z:msup><z:mo>,</z:mo><z:mi>y</z:mi></z:mrow></z:mfenced></z:mrow><z:mrow><z:mi>l</z:mi></z:mrow></z:msup><z:mo>⊆</z:mo><z:msup><z:mrow><z:mfenced open="{" close="}" separators="|"><z:mrow><z:msub><z:mrow><z:mi>y</z:mi></z:mrow><z:mrow><z:mi>β</z:mi></z:mrow></z:msub></z:mrow></z:mfenced></z:mrow><z:mrow><z:mi>l</z:mi></z:mrow></z:msup></z:math> whenever <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" id="M11"><kb:mi>α</kb:mi><kb:mo>≥</kb:mo><kb:msub><kb:mrow><kb:mi>α</kb:mi></kb:mrow><kb:mrow><kb:mn>0</kb:mn></kb:mrow></kb:msub></kb:math> . The emergence of a broader convergence, stemming from the recognition of more ordered vector spaces compared to lattice vector spaces, has prompted an expansion and broadening of discussions surrounding lattices to encompass additional spaces. We delve into studying the properties of this convergence and explore its relationships with other established order convergence. In every ordered vector space, we demonstrate that under certain conditions, every <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" id="M12"><mb:mi>u</mb:mi><mb:mi>o</mb:mi></mb:math> -convergent net implies <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" id="M13"><ob:mi>u</ob:mi><ob:mi>o</ob:mi></ob:math> -Cauchy, and vice versa. Let <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" id="M14"><qb:mi>X</qb:mi></qb:math> be an order dense subspace of the directed ordered vector space <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" id="M15"><sb:mi>Y</sb:mi></sb:math> . If <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" id="M16"><ub:mi>J</ub:mi><ub:mo>⊆</ub:mo><ub:mi>Y</ub:mi></ub:math> is a <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" id="M17"><wb:mi>u</wb:mi><wb:mi>o</wb:mi></wb:math> -band in <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" id="M18"><yb:mi>Y</yb:mi></yb:math> , then we establish that <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" id="M19"><ac:mi>J</ac:mi><ac:mo>∩</ac:mo><ac:mi>X</ac:mi></ac:math> is a <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" id="M20"><cc:mi>u</cc:mi><cc:mi>o</cc:mi></cc:math> -band in <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" id="M21"><ec:mi>X</ec:mi></ec:math> .