<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {ORD}_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">ORD</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> be the semigroup consisting of all oriented and order-decreasing full transformations on the finite chain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(X_{n}=\{ 1&lt;\cdots &lt;n \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>&lt;</mo> <mo>⋯</mo> <mo>&lt;</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1\le r\le n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, let <Equation ID="Equ5"> <EquationSource Format="TEX">\(\begin{aligned} \mathcal {ORD}(n,r) =\{\alpha \in \mathcal {ORD}_{n}\, :\, |\mathrm {im\, }(\alpha )|\le r\}. \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="script">ORD</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mi>α</mi> <mo>∈</mo> <msub> <mi mathvariant="script">ORD</mi> <mi>n</mi> </msub> <mspace width="0.166667em" /> <mo>:</mo> <mspace width="0.166667em" /> <mo stretchy="false">|</mo> <mrow> <mi mathvariant="normal">im</mi> <mspace width="0.166667em" /> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>r</mi> <mo stretchy="false">}</mo> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we determine the cardinality of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {ORD}(n,r)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">ORD</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the number of nilpotent elements of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal {ORD}(n,r)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">ORD</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we find a minimal generating set and the rank of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathcal {ORD}(n,r)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">ORD</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and moreover, we characterize all maximal subsemigroups of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathcal {ORD}(n,r)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">ORD</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for each <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(3\le r\le n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>