<p>In this paper, we study ordered ternary semigroups with an emphasis on those whose right ideals are linearly ordered by inclusion - these are called right chain ordered ternary semigroups and are a natural generalization of right chain semigroups. A&#xa0;prime segment of an ordered ternary semigroup <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( S \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation> is a pair <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( A \varsubsetneq B \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mspace width="0.333333em" /> <mpadded voffset="-0.45ex"> <mo mathsize="small">/</mo> </mpadded> <mspace width="-0.65em" /> <mo>⊆</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> of neighbouring completely prime ideals of&#xa0;<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( S \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> </InlineEquation>. We obtain a classification of prime segments of right chain ordered ternary semigroups as simple (i.e., there are no further ideals between <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( A \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( B \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>B</mi> </math></EquationSource> </InlineEquation>), Archimedean (i.e., for any <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( b \in B \setminus A \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> the ideal <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\( I \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>I</mi> </math></EquationSource> </InlineEquation> generated by <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\( b \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>b</mi> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\( \bigcap _{n \in \mathbb {N}}(I^n] = A \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>⋂</mo> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>I</mi> <mi>n</mi> </msup> <mo stretchy="false">]</mo> </mrow> <mo>=</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>), exceptional (i.e., there exists a prime ideal lying properly between <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\( A \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\( B \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>B</mi> </math></EquationSource> </InlineEquation>), or supplementary (i.e., there exists the smallest ideal among the ideals lying properly between <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\( A \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\( B \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>B</mi> </math></EquationSource> </InlineEquation>). A key role in achieving this classification is played by two constructions of ideals in any ordered ternary semigroup, which in the case of right chain ordered ternary semigroups yield prime and completely prime ideals with certain special properties.</p>