<p>We introduce a class of left ideals (and subalgebras) of nest algebras determined by totally ordered families of partial isometries on a complex Hilbert space <i>H</i>. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {E}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation> be a family of partial isometries that is totally ordered in the Halmos–McLaughlin ordering, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({{\mathcal {A}}}_{\mathcal {E}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="script">E</mi> </msub> </math></EquationSource> </InlineEquation> be the subset of operators in <i>B</i>(<i>H</i>) which, for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(E\in \mathcal {E}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>∈</mo> <mi mathvariant="script">E</mi> </mrow> </math></EquationSource> </InlineEquation>, map the initial space of <i>E</i> to the final space of <i>E</i>. We show that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({{\mathcal {A}}}_{\mathcal {E}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="script">E</mi> </msub> </math></EquationSource> </InlineEquation> is a subalgebra of <i>B</i>(<i>H</i>) if and only if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\({{\mathcal {A}}}_{\mathcal {E}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="script">E</mi> </msub> </math></EquationSource> </InlineEquation> is a left ideal of a certain nest algebra, and if so, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathcal {E}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation> consists of power partial isometries, except possibly for its supremum <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\vee \mathcal {E}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∨</mo> <mi mathvariant="script">E</mi> </mrow> </math></EquationSource> </InlineEquation>, in which case the range <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\operatorname {ran}(\vee \mathcal {E})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>ran</mo> <mo stretchy="false">(</mo> <mo>∨</mo> <mi mathvariant="script">E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is <i>H</i>. It is also shown that any left ideal <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\({{\mathcal {A}}}_{\mathcal {E}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="script">E</mi> </msub> </math></EquationSource> </InlineEquation> is decomposable and that the subset of finite-rank operators in its closed unit ball is strongly dense in the ball. Necessary and sufficient conditions to solve <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(Tx=y\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(T^*x=y\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\({{\mathcal {A}}}_{\mathcal {E}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="script">E</mi> </msub> </math></EquationSource> </InlineEquation> are given.</p>