Theresa May has left Northern Ireland without a deal that would have seen the restoration of power-sharing government in the region.

The prime minister and the Irish taoiseach, Leo Varadkar, had travelled to Belfast on Monday hoping that the main parties represented in the Stormont assembly would reach a compromise ending 13 months of political deadlock.

But for the second time in less than a year the Democratic Unionist party kept May waiting. The party held out for a deal over several weeks last summer with the prime minister and the Tories that kept May in Downing Street after the general election. That eventually resulted in the DUP’s confidence and supply arrangement with the Tories, which still shores up the minority Conservative government.

On Monday the DUP was holding out on another deal May was seeking, this time involving compromises over Sinn Féin’s demands for an Irish language act that would lead to the resurrection of a cross-community government in Belfast.

DUP sources said there was still grave disquiet over any deal that could be perceived as the party supporting a stand-alone Irish language act that would put Gaelic on the same legal status as English throughout Northern Ireland.

Quick Guide Languages in Northern Ireland Show Key facts about Irish and Ulster Scots Any deal to restore a cross-community power-sharing government in Northern Ireland will be made or broken on the issue of language. One of Sinn Féin's core demands has been the creation of an Irish language act to give Gaelic the same legal status as English. Unionists, and in particular the Democratic Unionist party, have resisted this, arguing that it would not only be too costly but would lead to daily culture wars over language on street signs and the way court cases are conducted, and would also "hollow out" British identity in the region. One way around this is to create a parallel act that would give special status to Ulster Scots and "Orange" culture, which the DUP could claim as its own victory from the talks. Here are some core facts about language and cultural identity issues that have affected the negotiations:

• In the 2011 census, 179,000 people – or 11% of the population of the region – claimed some knowledge of Irish. But when it came to Irish as their main language, this fell to 4,045. However, Gaelic language activists point out that more than 6,000 children are now studying at Irish-first schools and the sector is growing. • From the same census figures, more than 140,000 people say they speak Ulster Scots; the majority of them, though, are not exclusively from the unionist or Protestant population. However, the definition of Ulster Scots itself is contentious as many experts dispute that it is a language in its own right, instead calling it a distinct dialect. • A separate culture act could also be contentious if it were to guarantee the right of Orange parades to march on Northern Irish streets given the territorial disputes connected to loyalist demonstrations in some areas in the 1990s and early 2000s.



But despite returning home with no breakthrough May and Varadkar tried to paint an optimistic picture of the talks that they had jointly held at Stormont.

May said she believed a power-sharing executive could be “up and running very soon” and insisted that the main parties – the DUP and Sinn Féin – were actually close to a compromise deal.

After holding talks with the five main parties represented in the Stormont Assembly, May said: “While some differences remain, I believe it is possible to see the basis of an agreement.”

May said it was time for a “final push” towards an overall settlement.

Vardakar also attempted to paint a positive picture of the talks, stating that he was “very hopeful” an agreement could be reached between the DUP and Sinn Féin later this week.

All the main protagonists in these latest talks were also keen to talk up the chances of a positive outcome.

Sinn Féin’s new leader, Mary Lou McDonald, led her party’s delegation into meetings with May and Varadkar.

While McDonald confirmed Sinn Féin did not have the opportunity to meet directly with the DUP, she stressed that both parties would be holding talks “to resolve the outstanding issues”.

She added: “We are not exactly there just yet, but there is nothing insurmountable if there is the political will to reach an agreement.”

Arlene Foster, the former first minister of Northern Ireland and DUP leader, said some “very good progress” had been made in the talks so far.

While Foster said there was no deal, yet she wanted an agreement where there was a need “to respect all languages and cultures in Northern Ireland and not allow one to dominate over another”.

Overall Foster described the tone of the discussions as “very good”.

Mary Lou McDonald, Sinn Féin president: ‘There is nothing insurmountable if there is the political will to reach an agreement.’ Photograph: Paul Faith/AFP/Getty Images

Through 13 months of negotiations since the power-sharing government broke down, Sinn Féin’s core has been fixed on the drafting of an Irish language act. According to the talks source, there could now be three separate acts dealing with language and cultural rights.

One would secure the rights of Irish language speakers but there would also be legislation to enforce the rights of Ulster Scots speakers and a general cultural act designed in particularly around the promotion of Orange/Protestant culture.

The DUP is currently trying to sell this “three-act” compromise to its party grassroots.

Following their meeting with May the Ulster Unionist party said a Sinn Féin-DUP agreement was “not over the line” and that the other parties had been effectively excluded from all the main negotiations.

The initial reason for the collapse of the last power-sharing executive in Belfast was a controversial green energy scheme whose multimillion pound costs to taxpayers had spiralled out of control.

Because Foster and the DUP had championed the renewable energy initiative their key partners in that government had asked her to step aside temporarily from the first minister’s office so that an inquiry could held into the scheme.

When Foster refused to stand down the deputy first minister, the late Martin McGuinness, resigned from his post. Under the interlocking rules of power sharing if one main representative from the two divided communities in Northern Ireland resigns from office then the whole devolved edifice collapses.