I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure with $0$ of an $E_{\infty}$ ring space or, essentially equivalently, the multiplicative structure with $0$ of an $E_{\infty}$ ring spectrum. $E_{\infty}$ ring spaces with $0$ come with two basepoints, say $0$ and $1$, of which only $0$ should be thought of as truly the basepoint. $E_{\infty}$ ring spaces with zero then give the multiplicative structure of $E_{\infty}$ ring spaces, in which one has two interrelated $E_{\infty}$ space structures, one "additive'' and one "multiplicative'' but with $0$. The $0$ in the multiplicative structure allows one to form two monads in the same category, the category of based spaces, which are interrelated in the sense prescribed in a beautiful 1969 paper "Distributive laws" of Beck. Beck's theory is recalled in section 15 of "The construction of $E_{\infty}$ ring spaces from bipermutative categories". That paper and its companions "What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra" and "What are $E_{\infty}$ ring spaces good for?'' date to 2009. They modernize the theory first developed in the 1977 Springer volume "$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra'' that has already been referred to. It was first proven there that grouplike $E_{\infty}$ ring spaces, those for which $\pi_0$ is a group under addition and therefore a ring, are equivalent to connective $E_{\infty}$ ring spectra. This and the whole of infinite loop space theory, additive, multiplicative, and especially equivariant are being reworked in a paper provisionally titled ``Group completions and the homotopical monadicity theorem'', by Hana Jia Kong, Foling Zou, and myself.

I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure of an $E_{\infty}$ ring space or, essentially equivalently, the multiplicative structure of an $E_{\infty}$ ring spectrum. $E_{\infty}$ ring spaces or even just $E_{\infty}$ spaces with $0$, come with two basepoints, say $0$ and $1$, of which only $0$ should be thought of as truly the basepoint. $E_{\infty}$ spaces with zero then give the multiplicative structure of $E_{\infty}$ ring spaces, in which one has two interrelated $E_{\infty}$ space structures, one "additive'' and one "multiplicative'' but with $0$. The $0$ in the multiplicative structure allows one to form two monads in the same category, the category of based spaces, which are interrelated in the sense prescribed in a beautiful 1969 paper "Distributive laws" of Beck. Beck's theory is recalled in section 15 of "The construction of $E_{\infty}$ ring spaces from bipermutative categories". That paper and its companions "What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra" and "What are $E_{\infty}$ ring spaces good for?'' date to 2009. They modernize the theory first developed in the 1977 Springer volume "$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra'' that has already been referred to. It was first proven there that grouplike $E_{\infty}$ ring spaces, those for which $\pi_0$ is a group under addition and therefore a commutative ring, are equivalent to connective $E_{\infty}$ ring spectra. This and the whole of infinite loop space theory, additive, multiplicative, and especially equivariant are being reworked in a paper provisionally titled ``Group completions and the homotopical monadicity theorem'', by Hana Jia Kong, Foling Zou, and myself.

I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure with $0$ of an $E_{\infty}$ ring space or, essentially equivalently, the multiplicative structure with $0$ of an $E_{\infty}$ ring spectrum. $E_{\infty}$ ring spaces with $0$ come with two basepoints, say $0$ and $1$, of which only $0$ should be thought of as truly the basepoint. $E_{\infty}$ ring spaces with zero then give the multiplicative structure of $E_{\infty}$ ring spaces, in which one has two interrelated $E_{\infty}$ space structures, one "additive'' and one "multiplicative'' but with $0$. The $0$ in the multiplicative structure allows one to form two monads in the same category, the category of based spaces, which are interrelated in the sense prescribed in a beautiful 1969 paper "Distributive laws" of Beck. Beck's theory is recalled in section 15 of \url{http://www.math.uchicago.edu/~may/PAPERS/Final2.pdf}, "The construction of $E_{\infty}$ ring spaces from bipermutative categories". That paper and its companions \url{http://www.math.uchicago.edu/~may/PAPERS/Final1.pdf} "What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra" and \url{http://www.math.uchicago.edu/~may/PAPERS/Final3.pdf} "What are $E_{\infty}$ ring spaces good for?'' date to 2009. They modernize the theory first developed in the 1977 Springer volume \url{http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf} "$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra'' that has already been referred to. It was first proven there that grouplike $E_{\infty}$ ring spaces, those for which $\pi_0$ is a group under addition and therefore a ring, are equivalent to connective $E_{\infty}$ ring spectra. This and the whole of infinite loop space theory, additive, multiplicative, and especially equivariant are being reworked in a paper provisionally titled ``Group completions and the homotopical monadicity theorem'', by Hana Jia Kong, Foling Zou, and myself.

I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure with $0$ of an $E_{\infty}$ ring space or, essentially equivalently, the multiplicative structure with $0$ of an $E_{\infty}$ ring spectrum. $E_{\infty}$ ring spaces with $0$ come with two basepoints, say $0$ and $1$, of which only $0$ should be thought of as truly the basepoint. $E_{\infty}$ ring spaces with zero then give the multiplicative structure of $E_{\infty}$ ring spaces, in which one has two interrelated $E_{\infty}$ space structures, one "additive'' and one "multiplicative'' but with $0$. The $0$ in the multiplicative structure allows one to form two monads in the same category, the category of based spaces, which are interrelated in the sense prescribed in a beautiful 1969 paper "Distributive laws" of Beck. Beck's theory is recalled in section 15 of "The construction of $E_{\infty}$ ring spaces from bipermutative categories". That paper and its companions "What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra" and "What are $E_{\infty}$ ring spaces good for?'' date to 2009. They modernize the theory first developed in the 1977 Springer volume "$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra'' that has already been referred to. It was first proven there that grouplike $E_{\infty}$ ring spaces, those for which $\pi_0$ is a group under addition and therefore a ring, are equivalent to connective $E_{\infty}$ ring spectra. This and the whole of infinite loop space theory, additive, multiplicative, and especially equivariant are being reworked in a paper provisionally titled ``Group completions and the homotopical monadicity theorem'', by Hana Jia Kong, Foling Zou, and myself.