# Model Categories and Their Localizations by Philip S. Hirschhorn

By Philip S. Hirschhorn

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If V in addition being a set neutrosophic modulo integer vector space over S satisfies the condition, that for every pair v, u V, u + v and v + u V; then we call V to be a set neutrosophic modulo integer linear algebra over S. We illustrate this by some simple examples. 23: Let V = {0, I, 2I, 3I, 4I, 5I, 6I, 7I, 8I, 9I, 10I} N(Z11) and S = {0, 1, I, 5, 3, 2I, 6I, 8I} N(Z11). V is a set neutrosophic modulo integer linear algebra over S. 24: Let V = {0, 2I, 4I, 6I, 8I, 10I, 12I, 14I, 16I} N(Z18), S = {0, 1, 2, 4, 8, 2I, 6I, 10I}.

We say V is a neutrosophic-neutrosophic integer set vector space over S N(Z) (S Z) if sQi = Qi s V for every Qi V and s S. We shall for easy representation write neutrosophicneutrosophic integer vector space as n-n integer set vector space. We now illustrate this new structure by some examples. 1: Let V = {0, 1 + (2n – 1)I | n = 1, 2, …, f} N(Z). V is a n-n integer set vector space over S = {0, 1 + I, 1} N(Z). 2: Let V = {I, 2I, 5I, 7I, 0, 8I, 27I} N(Z). V is a n-n integer set vector space over S = {0, I} N(Z).

So PN(Q) hereafter will be known as pure set neutrosophic rational numbers. 1: Let V N(Q) (PN(Q)) be a proper subset of N(Q) or V contains elements from N(Q) (PN(Q)) (V Q). Let S N(Q) be a proper subset of N(Q). We say V is a mixed (pure) set neutrosophic rational vector space over S if sQ V for every s S and Q V. 1: Let 19 19I 27 27I 2 2I , , V = ® , 0, 2 2 5 5 ¯7 7 17I, 48 – 48I, 28 28I 47I ½ , ¾ N(Q). 13 13 5 ¿ Take S = {0, 1, 11 11I , 1 – I} N(Q). 7 7 It is easily verified V is a pure set neutrosophic rational vector space over the set S.