You can use my \Cen
macro, see https://tex.stackexchange.com/a/209732/4427
\documentclass[a4paper,12pt,reqno]{amsart}
\makeatletter
\newcommand{\Cen}[2]{%
\ifmeasuring@
#2%
\else
\makebox[\ifcase\expandafter #1\maxcolumn@widths\fi]{$\displaystyle#2$}%
\fi
}
\makeatother
\begin{document}
\begingroup\renewcommand{\theequation}{\arabic{equation}a-b}
\begin{flalign}
&\text{Gas injection:} &
&\Cen{4}{\xi = Y_i,} &
&\Cen{6}{\sigma = \left( \frac{\bar{s}_{\mathrm{air}}-\bar{s}_{i}}{\bar{c}_p} \right)\xi,} &&
\\[1ex]
&\text{Heat addition:} &
&\Cen{4}{\xi = 0,} &
&\Cen{6}{\sigma = T^\prime/\bar{T},} &&
\end{flalign}
\endgroup
\end{document}

I'd also consider a standard alignment:
\begingroup\renewcommand{\theequation}{\arabic{equation}a-b}
\begin{flalign}
&\text{Gas injection:} &
\xi &= Y_i, &
\sigma &= \left( \frac{\bar{s}_{\mathrm{air}}-\bar{s}_{i}}{\bar{c}_p} \right)\xi, &&
\\[1ex]
&\text{Heat addition:} &
\xi &= 0, &
\sigma &= T^\prime/\bar{T}, &&
\end{flalign}
\endgroup

You can label the equations separately, but you have to explicitly give the subequation letter. I also implemented math mode for the subequation letters (although I'd not use it myself).
\documentclass[a4paper,12pt,reqno]{amsart}
\makeatletter
\newcommand{\Cen}[2]{%
\ifmeasuring@
#2%
\else
\makebox[\ifcase\expandafter #1\maxcolumn@widths\fi]{$\displaystyle#2$}%
\fi
}
\let\extra@label\label
\newcounter{equationtemp}
\newcommand{\extralabel}[2]{%
\begingroup
\setcounter{equationtemp}{\value{equation}}%
\renewcommand{\theequationtemp}{\arabic{equationtemp}$#1$}%
\refstepcounter{equationtemp}%
\extra@label{#2}%
\endgroup
}
\makeatother
\begin{document}
\begingroup\renewcommand{\theequation}{\arabic{equation}$a$-$b$}
\begin{flalign}
\label{first}
&\text{Gas injection:} &
&\Cen{4}{\xi = Y_i,} \extralabel{a}{one} &
&\Cen{6}{\sigma = \left( \frac{\bar{s}_{\mathrm{air}}-\bar{s}_{i}}{\bar{c}_p} \right)\xi,}
\extralabel{b}{two} &&
\\[1ex]
\label{second}
&\text{Heat addition:} &
&\Cen{4}{\xi = 0,} \extralabel{a}{three} &
&\Cen{6}{\sigma = T^\prime/\bar{T},} \extralabel{b}{four} &&
\end{flalign}
\endgroup
\ref{one} and \ref{two}; \ref{three} and \ref{four}; \ref{first} and \ref{second}.
\eqref{one} and \eqref{two}; \eqref{three} and \eqref{four}; \eqref{first} and \eqref{second}.
\end{document}
