Sources that prohibit Muslim prayer rugs as decorations for synagogues

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First an introduction. I see a weakening of some elements in the Jewish community towards the religion of Islam, so this is a timely subject
For example
http://www.theyeshivaworld.com/news/headlines-breaking-stories/371240/canadian-mosque-burned-down-local-synagogue-offers-a-place-to-pray.html

In Canada, the Masjid al-Salaam mosque, the only mosque in Peterborough, Ontario was burned down in an arson attack the day after the Paris attacks on November 14.

Congregants of that mosque are now praying in a nearby synagogue.

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Someone had thrown a Molotov cocktail at their building and the damages have been estimated at about €56,000.

Following this incident, the leaders of the city’s only synagogue, Beth Israel, offered to have the mosque hold their services at theeir synagogue so that they would have somewhere to pray. “We live in a small town and they are our neighbors, so it seemed only natural to welcome them,” says the synagogue’s president, Larry Gillman.

Last Friday, about 150 Muslims went to pray at Beth Israel.

Kenzu Abdella, the president of the Kawartha Muslim Religious Association (KMRA), which runs the mosque said,

“This is the first time the mosque has been attacked. We are shocked and find this very worrisome. However, we have received an outpouring of support. The city authorities, the synagogue and several churches all offered us spaces in which to continue our activities.

“Since the fire, we’ve prayed at the United Church of Canada on Friday November 20, at the synagogue on Friday November 27, and we’re going to go to an Anglican church this Friday. The rest of the week, we pray at the home of one of our members, who lives near the mosque.

“When Larry Gillman invited us to the synagogue, I first wondered how we would organise the prayer, since we need rugs, we need to pray in a certain direction, etc. We ended up praying in a room on the first floor and it went very well, thanks to the people who run the synagogue. We’re very grateful.

“After the prayer, we held our annual dinner at the synagogue. But this time, we cooked a vegetarian dinner in order not to go against the customs of the synagogue (where kosher meals are served). This was an occasion for the two communities to get to know each other better and to start a dialogue. I think that’s the best way to combat radicalism and hate.”

Joe Teichman,the general manager of Beth Israel synagogue, said,

“During the dinner we shared on Friday, there were about 80 Muslim congregants and 15 Jewish ones. We talked about all sorts of things, and realised we had a lot in common. For example, many words in Hebrew and in Arabic share the same root. Since we have a big kitchen at the synagogue, the Muslim congregants suggested we might organise cooking classes together in the future. We’ve already invited them back for another Friday prayer in December.

“The different religious communities of Peterborough were already in contact with each other prior to this. For example, for the past decade, we’ve held an interfaith film festival where Christians, Jews, and Muslims watch movies and discuss them together.”

Reconstruction of the mosque began three days after the fire, and is expected to be finished by the end of the month. More than C$110,000 (about €76,000) has been raised in just 48 hours on a fundraising site to cover the costs.

Following the Paris attacks, numerous attacks against Muslims were reported in Canada, and notably in Ontario.
There's even a video about it, where you see them on their prayer rugs in the synagogue https://youtu.be/Ea2HUQR5q78

In addition read the article
Chief Rabbi Of England Backs Teaching Islam In Jewish Schools
http://www.theyeshivaworld.com/news/headlines-breaking-stories/371364/chief-rabbi-of-england-backs-teaching-islam-in-jewish-schools.html

I will try in a few posts to bring some sources against this liberal practice of Beth Israel
By the way I have no information one way or the other if Beth Israel is an Orthodox synagogue, but the Chief Rabbi of England is definitely Orthodox

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This what the Rosh (Rabbi Asher ben Yechiel) has to say about this. Rosh is one of the 3 basic sources used by the Shulchan Aruch to decide Jewish Law
First the original Hebrew and then a translation
שו"ת הרא"ש כלל ה סימן ב

מה ששאלת בענין מחצלת קטנה הנקראת סגאדה בלשון ערבי, שדרך ישמעאלים להתפלל עליה ויש בה כמו צורת משקל שחור, אם מותר לתלותה בבית הכנסת לצד ההיכל, אחד מכאן ואחד מכאן, ולהתפלל כנגדה. ואף על פי שחלילה וחס לשום אדם אחד מישראל להתפלל בשום כונה לדבר אסור. דבר זה שאלתי בעבורו וחקרתי עליו, ונתברר לי שנהגו איסור בכאן, בטוליטלא, להניח מחצלת כזו בבית הכנסת כדי לישב עליה, כל שכן לתלותה בצד ההיכל. לפי שאומרים שאותה צורה שחורה העשויה בה היא צורת המקום שהולכים לחוג לשם בארצם. וגם יש אומרים שיש בה צורת מרקוליס, ודרכם להתפלל עליה ולכרוע עליה' בתפלתם. ועוד אמרו לי: מפני זה קורין למחצלת סגאדה, לפי שסוגדין עליה. וכיון שכן, נראה לי שאסור לתלותה בבית הכנסת כלל, כל שכן בצדי ההיכל, וצריך להוציאה מבית הכנסת כדי שלא ישאר לה שום זכר בבית הכנסת. כי מה לנו להניח בבית הכנסת שלנו דבר כזה, שהוא מוכן ומזומן לתפלתם? ואין ראוי לשום אדם מישראל לערער על זה, כדי להניח מחצלת כזו בבית הכנסת; וכל העובר על זה הוא נוטה מדרך ישרה לדרך רעה וראוי ליסרו. הנראה בעיני צויתי לכתוב וחתמתי עליו. אשר בן ה"ר יחיאל ז"ל.

What you asked about the matter of the small prayer rug called Segada in Arabic, that it is the manner of the Muslims to pray  on it and it has some type of a form of a black weight (or perhaps scale?) on it, is it permissible to suspend it in a synagogue by the side of the place where the Torah scrolls are stored, one on this side and one on the other, and to pray opposite it, even though, heaven forbid there is no one from Israel that has any intent for a prohibition. This matter I was asked to deal with it and I investigated it and it became clear to me that they were accustomed to prohibit it, here in (the place called) Tultila to put such a rug (or mat) like this in the synagogue and sit on it, and all the more so to hang it by the side of the heichal (where the Torah scrolls are stored), because they say that the black form found on it,  is like the form of the place that they go to celebrate their festival in their land. Also there are those that say that it has the shape of {the idol} Markolis, and their manner is to pray on it and bend down in their prayer. Further they tell me that they call this rug (or mat) Sagada because they bow on it. Now since this is so, it appears to me that it is forbidden to hang it in a synagogue entirely, all the more so by the side of the Heichal, and one has to remove it from the synagogue so that there doesn't remain any rememberance of it in the synagogue, for what cause do we have to leave in a synagogue something like this, that is prepared for and ready for their prayers? And it is not fitting for any person of Israel to contest this, in order to leave a rug (or mat) like this in the synagogue; and all that transgress this depart from the upright way to an evil way and it is fitting to chastise him.

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Rabbi Yosef Karo in his responsa Avkat Rochel, said that both Rambam and Rosh would oppose using the Muslim prayer rugs as decorations in synagogues, although for different reasons.
שו"ת אבקת רוכל סימן סג

וכבר נמצא להרא"ש ז"ל שנשאל על מחצלת שדרך הישמעאלים להתפלל עליה אם מותר לתלותה בב"ה אע"פי שחלילה וחס לשום אחד מישראל להתפלל עליה בשום כוונה לשום דבר איסור והשיב אסור לתלותה בבית הכנסת כלל ואין ראוי לשום אדם הישראל /מישראל/ לערער ע"ז וכל העובר על זה הוא נוטה מדרך ישרה לרעה וראוי לייסרו עכ"ל ומטעם אחר נמי אסור לדעת הרמב"ם שכתב בתשובה הבגדים המצויירים אף על פי שאינן בולטות אין נכון להתפלל כנגדה כדי שלא יהיה מביט בציורים ההם ולא יכוין בתפלתו

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