Who are the “elect” in Matthew 24:22 and what is the purpose of shortening the days for their sake?

Just as Our Lord’s Great Prophecy (Matthew 24) applies now to the close of the Gospel Age, it also applied to the close of the Jewish Age.

Jerusalem was sieged by the desolating Roman armies, bringing about a time of intense distress on the people. The famine and internal anarchy of this bleak period are legend. An opportunity for flight came in A.D. 68 when Roman forces returned to Rome. However, flight required faith and a willingness to suffer the pains of a refugee.  Flight was the last and only remedy for the Jews. The flight was more difficult for those with children and those during the winter season (verses 19-20).  Those were days of great distress and affliction. Had not the Lord intervened, the opportunity for this difficult flight would not have come. But for the “elect’s sake” [the faithful Jews] (verse 22) he did intervene, so that they were not swept away in the carnage that followed.

If we relate these same verses (Matthew 24:19-22) to the end of the Gospel Age, we can see that the Lord is the refuge for the true Christian fleeing Christendom. The flight will be difficult. Pray that it be not in the wintertime but in the more favorable summer time of harvest. The original Greek text (Matthew 24:22) in Wilson’s Emphatic Diaglott, reads, “And unless those days were cut short [cut off, amputated, Greek koloboo] No One could survive; but on account of the chosen, those days will be limited.” This translation implies that the Lord will cut the forces gathering for the last debacle.  He will “hold back the four winds” to allow the saints living at the very end of this age to make their calling and election, until their sealing is complete.

 Since we interpret Matthew 24:22 as meaning that there are still some “foot” members of the church on the scene during the very last days of the harvest who are in the process of working out their calling and election, the Lord will hold back the forces of Armageddon for their sakes or, on account of them, so that they can complete what they have started.

Who are the four horns?

We will assume that you are referring to Zech 1:18-21.

Zech 1:18-21  "Then lifted I up mine eyes, and saw, and behold four horns. And I said unto the angel that talked with me, What be these? And he answered me, These are the horns which have scattered Judah, Israel, and Jerusalem. And the LORD shewed me four carpenters. Then said I, What come these to do? And he spake, saying, These are the horns which have scattered Judah, so that no man did lift up his head: but these are come to fray them, to cast out the horns of the Gentiles, which lifted up their horn over the land of Judah to scatter it."

In verses 18-21, Zechariah beheld four horns and four carpenters. In the Scriptures, a “horn” is a symbol of power (whether on an altar, on an animal, or used figuratively), and a “carpenter” is a symbol of one who builds up. Therefore, the four horns in this vision are four powers, but what powers? The effect of these powers was to scatter (punish) Judah, Israel, and Jerusalem. The implication in verse 21 is that Gentile powers did the scattering. 
 
In this particular context, the sequence is Judah, Israel, and Jerusalem (the city). Judah, the southern kingdom was scattered by the Gentile powers of Egypt and Babylon. With Israel, the northern kingdom, the scattering Gentile powers were Assyria and Syria. But since the Old Testament prophecies are written for the Christian’s admonition, it is likely the primary application pertains to a greater picture with a different starting point than either Egypt or Assyria.
 
The four universal empires listed in Scripture are Babylon, Media-Persia, Greece, and Rome. Consider King Nebuchadnezzar, an enemy of Israel, who was called a “lion” that would come down from the north and scatter Israel (Jer. 4:7). God also called him “my servant” (Jer. 25:9). Nebuchadnezzar was permitted to come down against Israel to destroy Jerusalem and Solomon’s Temple because of the nation’s previous disobedience and waywardness. And even when Cyrus, the king of Persia, allowed the Israelites to go back to their homeland, they were a subject people.  Therefore, these four horns in Zechariah most likely picture the four major Gentile powers, or empires, starting with Babylon and corresponding with the image in the second chapter of Daniel. After Babylon was Media-Persia, then Greece, then Rome.