Let K be an arbitrary field. Let ( q n ) ({q_n}) be a sequence of positive integers, and let there be given a family { Ψ n m | n ≥ m } \{ {\Psi _{nm}}|n \geq m\} of unital K-monomorphisms Ψ n m : T q m ( K ) → T q n ( K ) {\Psi _{nm}}:{T_{{q_m}}}(K) \to {T_{{q_n}}}(K) such that Ψ n p Ψ p m = Ψ n m {\Psi _{np}}{\Psi _{pm}} = {\Psi _{nm}} whenever m ≤ n m \leq n , where T q n ( K ) {T_{{q_n}}}(K) is the K-algebra of all q n × q n {q_n} \times {q_n} upper triangular matrices over K. A triangular UHF (TUHF) K-algebra is any K-algebra that is K-isomorphic to an algebraic inductive limit of the form T = lim → ( T q n ( K ) ; Ψ n m ) \mathcal {T} = \lim \limits _ \to ({T_{{q_n}}}(K);{\Psi _{nm}}) . The first result of the paper is that if the embeddings Ψ n m {\Psi _{nm}} satisfy certain natural dimensionality conditions and if S = lim → ( T p n ( K ) ; Φ n m ) \mathcal {S} = \lim \limits _ \to ({T_{{p_n}}}(K);{\Phi _{nm}}) is an arbitrary TUHF K-algebra, then S \mathcal {S} is K-isomorphic to T \mathcal {T} , only if the supernatural number, N [ ( p n ) ] N[({p_n})] , of ( q n ) ({q_n}) is less than or equal to the supernatural number, N [ ( p n ) ] N[({p_n})] , of ( p n ) ({p_n}) . Thus, if the embeddings Φ n m {\Phi _{nm}} also satisfy the above dimensionality conditions, then S \mathcal {S} is K-isomorphic to T \mathcal {T} , only if N [ ( p n ) ] = N [ ( q n ) ] N[({p_n})] = N[({q_n})] . The second result of the paper is a nontrivial "triangular" version of the fact that if p, q are positive integers, then there exists a unital K-monomorphism Φ : M q ( K ) → M p ( K ) \Phi :{M_q}(K) \to {M_p}(K) , only if q | p q|p . The first result of the paper depends directly on the second result.