  An infinite dimensional algebra denoted $\bar{\cal A}_q$ that is isomorphic
to a central extension of $U_q^+$ - the positive part of $U_q(\widehat{sl_2})$
- has been recently proposed by Paul Terwilliger. It provides an `alternating'
Poincar\'e-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis
built from positive root vectors. In this paper, a presentation of $\bar{\cal
A}_q$ in terms of a Freidel-Maillet type algebra is obtained. Using this
presentation: (a) finite dimensional tensor product representations for
$\bar{\cal A}_q$ are constructed; (b) explicit isomorphisms from $\bar{\cal
A}_q$ to certain Drinfeld type `alternating' subalgebras of
$U_q(\widehat{gl_2})$ are obtained; (c) the image in $U_q^+$ of all the
generators of $\bar{\cal A}_q$ in terms of Damiani's root vectors is obtained.
A new tensor product decomposition for $U_q(\widehat{sl_2})$ in terms of
Drinfeld type `alternating' subalgebras follows. The specialization
$q\rightarrow 1$ of $\bar{\cal A}_q$ is also introduced and studied in details.
In this case, a presentation is given as a non-standard Yang-Baxter algebra.
This paper is dedicated to Paul Terwilliger for his 65th birthday.
