Is the son of a half-sister regarded as a mahram?
Question

                My wife has a half-sister through her mother, i.e., from a different father, and this sister has a son who is twenty years old. My question is: is this son a mahram to my wife? Is it permissible for her to shake hands with him?
Praise be to Allah.The son of a full sister or a half-sister through the mother or father is one of a woman’s mahrams, because Allah, may He be exalted, says (interpretation of the meaning): “Forbidden to you (for marriage) are: your mothers, your daughters, your sisters, your fathers’ sisters, your mothers’ sisters, your brothers’ daughters, your sisters’ daughters” [an-Nisa’ 4:23]. 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
There are seven categories of mahrams according to the text and scholarly consensus, and none of the scholars differed concerning this matter.
End quote from ash-Sharh al-Mumti‘ (12/110) 
They are: 
1-mothers – which includes grandmothers on both the father’s and the mother’s sides  
2-daughters – which includes granddaughters
3-sisters – whether full sisters or half-sisters through the father or the mother
4-paternal aunts – which includes the paternal aunt of the father and the paternal aunt of the mother
5-maternal aunts – which includes the maternal aunt of the father and the maternal aunt of the mother
6-brothers’ daughters – which includes brothers’ granddaughters
7-sisters’ daughters – which includes sisters’ granddaughters.
Shaykh Siddeeq Hasan Khan (may Allah have mercy on him) said: at-Tahhaawi said: There is scholarly consensus on all of these mahrams and it is not permissible to marry any of them, according to scholarly consensus.
End quote from Nayl al-Maraam min Tafseer Ayaat al-Ahkaam (1/148); see also Ahkaam al-Qur’an by Ibn al-‘Arabi (2/259) 
Once it is established that the maternal aunt who is a full sister (of the mother) – or her half-sister through her mother or through her father – is a mahram, it is known that it is permissible to look at her as she appears in front of her other mahrams such as her brother, paternal uncle, maternal uncle and so on. It is also permissible for him to enter upon her even if there is no other mahram present, and to look at her, so long as the situation is within proper bounds, and there is no risk of temptation or suspicion, so that the Shaytaan will have no way to introduce any kind of doubt or suspicion into your mind. 
Allah, may He be exalted, says (interpretation of the meaning):  “…and not to reveal their adornment except to their husbands, their fathers, their husbands’ fathers, their sons, their husbands’ sons, their brothers or their brothers’ sons, or their sisters’ sons…” [an-Noor 24:31]. 
Ibn Katheer (may Allah have mercy on him) said: In the case of all these mahrams, it is permissible for a woman to appear before them with her adornment, but without making a wanton display.
End quote from Tafseer Ibn Katheer (10/220) 
Siddeeq Hasan Khan (may Allah have mercy on him) said:
It is permissible for women to show their adornment to these people, because they mix with them a great deal and there is no fear of temptation, because of the natural aversion that people have towards intimacy with close relatives.
End quote from Nayl al-Maraam min Tafseer Ayaat al-Ahkaam (1/397) 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: 
What about shaking hands with one’s maternal aunt? 
He replied: 
Shaking hands with one’s maternal aunt and other mahrams, such as one’s paternal aunt, brother’s daughter or sister’s daughter – and especially one’s daughter or mother – is permissible and there is nothing wrong with it, so long as there is no risk of temptation, and usually there is no such risk. The same applies to their equivalents among mahrams through breastfeeding: it is permissible to shake hands with them, so long as there is no risk of temptation, because the Prophet (blessings and peace of Allah be upon him) said: “What becomes mahram (forbidden for marriage) through breastfeeding is that which becomes mahram through blood ties.” 
End quote from Fataawa Noor ‘ala ad-Darb 
And Allah knows best.