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How can I change following cods to have better representation?

\documentclass[a4paper]{article}
\usepackage{amsmath}
\begin{document}

\begin{description}

\item $z_{i( j,l,m) }^{\perp }=\dfrac{( \mathrm{f}+1+2x_{jl}x_{jm}x_{lm}) z_{i( j,l,m) }-1}{\mathrm{f}+1} -%
\dfrac{2{ S_{\mathbf{z}_{( j,l,m) }}^{1\-- \mathrm{f}}}+(\mathrm{f}-1) {S_{\mathbf{z}_{( j,l,m) }}^{1\--2 \mathrm{f}}}}{%
2 \mathrm{f}( \mathrm{f}+1) }$,  if $i\in UC_{( j,l,m)}$,%

\item$ z_{i( j,l,m) }^{\perp }=\dfrac{2z_{i( j,l,m) }+{ S_{\mathbf{z}_{(
i,j,l,m) }}^{1\-- \mathrm{f}}} }{\mathrm{f}+1}-\dfrac{2{ S_{\mathbf{z}_{(
j,l,m) }}^{1\-- \mathrm{f}}} +( \mathrm{f}-1) { S_{\mathbf{z}_{(
j,l,m) }}^{1\--2 \mathrm{f}}} }{2 \mathrm{f}( \mathrm{f}+1) }$, if  $i\leq \mathrm{f}$, $i\notin UC _{( j,l,m) }$,%

\item$ z_{i( j,l,m) }^{\perp }=\dfrac{( \mathrm{f}+1-2x_{jl}x_{jm}x_{lm}) z_{i( j,l,m) }-1}{\mathrm{f}+1}+%
\dfrac{2{ S_{\mathbf{z}_{( j,l,m) }}^{1\-- \mathrm{f}}}-{(f+1) S_{\mathbf{z}_{( j,l,m) }}^{1\--2 \mathrm{f}}} }{%
2 \mathrm{f}}$, if $i\in BC_{( j,l,m) }$,%

\item$ z_{i( j,l,m) }^{\perp}=\dfrac{2z_{i( j,l,m) }+{ S_{\mathbf{z}_{(
i-\mathrm{f},j,l,m) }}^{\mathrm{f}+1\-- 2 \mathrm{f}}} }{\mathrm{f}+1}+\dfrac{2{ S_{\mathbf{z}_{(
j,l,m) }}^{1\-- \mathrm{f}}} -( \mathrm{f}+1) { S_{\mathbf{z}_{(
j,l,m) }}^{1\--2 \mathrm{f}}} }{2 \mathrm{f}( \mathrm{f}+1) }$, if $i\geq (f+1)$, $i\notin BC_{(j,l,m)}$.%
\end{description}

\end{document}

what is the replacement of description list? I can use the above items in a formula.

1
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I would suggest the use of multline* (or multline if you want equation numbers). Also I removed some comma's.

\documentclass[a4paper]{article}
\usepackage{amsmath}
\begin{document}

  \begin{multline*}
    z_{i( j,l,m) }^{\perp }=\dfrac{( \mathrm{f}+1+2x_{jl}x_{jm}x_{lm})
      z_{i( j,l,m) }-1}{\mathrm{f}+1} -\\
    \dfrac{2{ S_{\mathbf{z}_{( j,l,m) }}^{1\--
          \mathrm{f}}}+(\mathrm{f}-1) {S_{\mathbf{z}_{( j,l,m) }}^{1\--2
          \mathrm{f}}}}{%
      2 \mathrm{f}( \mathrm{f}+1) } \quad\text{if } i\in UC_{( j,l,m)}%
  \end{multline*}
  \begin{multline*}
     z_{i( j,l,m) }^{\perp }=\dfrac{2z_{i( j,l,m) }+{ S_{\mathbf{z}_{(
            i,j,l,m) }}^{1\-- \mathrm{f}}} }{\mathrm{f}+1}-\\
    \dfrac{2{S_{\mathbf{z}_{( j,l,m) }}^{1\-- \mathrm{f}}} +( \mathrm{f}-1) {
        S_{\mathbf{z}_{( j,l,m) }}^{1\--2 \mathrm{f}}} }{2 \mathrm{f}(
      \mathrm{f}+1) }\quad \text{if } i\leq \mathrm{f},
    i\notin UC _{( j,l,m) }%
  \end{multline*}
  \begin{multline*}
     z_{i( j,l,m) }^{\perp }=\dfrac{( \mathrm{f}+1-2x_{jl}x_{jm}x_{lm})
      z_{i( j,l,m) }-1}{\mathrm{f}+1}+\\
    \dfrac{2{ S_{\mathbf{z}_{( j,l,m) }}^{1\-- \mathrm{f}}}-{(f+1)
        S_{\mathbf{z}_{( j,l,m) }}^{1\--2 \mathrm{f}}} }{%
      2 \mathrm{f}}\quad \text{if } i\in BC_{( j,l,m) }
  \end{multline*}
  \begin{multline*}
     z_{i( j,l,m) }^{\perp}=\dfrac{2z_{i( j,l,m) }+{ S_{\mathbf{z}_{(
            i-\mathrm{f},j,l,m) }}^{\mathrm{f}+1\-- 2 \mathrm{f}}}
    }{\mathrm{f}+1}+\\\dfrac{2{ S_{\mathbf{z}_{( j,l,m) }}^{1\--
          \mathrm{f}}} -( \mathrm{f}+1) { S_{\mathbf{z}_{( j,l,m)
          }}^{1\--2 \mathrm{f}}} }{2 \mathrm{f}( \mathrm{f}+1)
    }\quad \text{if } i\geq (f+1), i\notin BC_{(j,l,m)}%
  \end{multline*}
\end{document}

enter image description here

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