Tens of thousands of people demonstrated across the country on Tuesday
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Mr Mursi has said the decree will be limited in scope, and his backers say the decree is needed to protect the gains of the revolution against a judiciary with ties to the Mubarak era.
Here people in Egypt share their views on the president's decision.
Salah Mustafa, Cairo,
I voted for Mursi but since his election he has been acting like a delegate of the Muslim Brotherhood to the presidential palace - and not as the president of all Egyptians.
He has been getting in bed with figures of the old regime, preparing a horrible draft constitution and now proclaiming himself a pharaoh.
The president's reassurances won't work, because all he is doing is anointing himself pharaoh one day, then saying he won't abuse this power on the next day.
We are not going to allow people to make the same mistakes we made in the past. President Nasser once promised to give power back to civilians.
Mursi must do two things. First, there must be checks and balances - he must cancel the decree that gave him extended powers. Secondly, he must reform the assembly that is writing the constitution so it represents everyone in the country.
I was in Tahrir Square during the revolution and I went there again on Tuesday.
It's unfortunate that my demands are still same - bread, freedom and social justice.
If Mursi doesn't get it, well, we have seen this movie before and all know where this is going to end.
Nabila Abdullah, Cairo
Not only do I agree, but a majority of people I know are in agreement and support President Mursi and what he did.
He did not give himself unchecked power.
He merely took unchecked power away from the judiciary - which is still full of personalities from the Mubarak era - until the constitution is done and there are systems in place to allow the country to move forward.
The president's declaration is an absolute necessity. We hear judges declaring that they will dissolve the upper house of parliament, or the constitution committee, weeks before their session is due and even before cases are properly presented or discussed.
Most importantly, by design this protection of the president's declarations is temporary - only until the constitution is written and congress is elected.
That is not dictatorship. What is a dictatorship is when judges overturn the people's elected institutions.
There are many polls taken from online media portals and social media pages that show an overwhelming majority support the decisions of the president.