On the Category of EQ-algebras

Abstract


 
 
 EQ-algebras were introduced by Nova ́k in [15] as an algebraic structure of truth values for fuzzy type theory (FFT). In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculate coprodut and pushout in a special case. Also, we construct a free EQ-algebra on a singleton.
 
 

\(n\)-Fold Filters of EQ-Algebras

“…Let E be a residuated EQ-algebra and F be a filter of E. Then F is an n-fold obstinate filter of E if and only if every filter of quotient algebra E/F is an n-fold obstinate filter of E/F . Proof: Let F be an n-fold obstinate filter of E and x ∈ E such that [x] ̸ = [1]. Then x / ∈ F and so there exists m ∈ N such that (¬(x n )) m ∈ F and so [(¬(x n )) m ] ∈ { [1]}.…”

“…But the binary relation introduced by prefilter is not a congruence relation. To learn more about EQ-algebras, the reader can consult [1,2,7,11,13,14]. Filter theory plays an important role in studying logical algebras.…”

Fuzzy n-fold obstinate and maximal (pre)filters of EQ-algebras

“…If t = 0, then µ t = E, and so µ t ∈ OF n (E). Suppose 0 ≤ t ≤ 1 2 and x, y / ∈ µ t . Then µ(x) < t and µ(y) < t, and so…”

“…For solving this problem, they added another condition to the definition of prefilter so filter of EQ-algebras is defined. To read more about EQ-algebras, the reader can read articles [1,3,12,20,22,24].…”

“…Then (1−χ {[1]} [x])∧(1−χ {[1]} [y]) ≤ χ {[1]} ([x n ] → [y])∧χ {[1]} ([y n ] → [x]). If [x] =[1] and [y] = [1], then µ(x) = µ(1) = 1 and µ(y) = µ(1) = 1 and so there exists m, k ∈ N such that µ(¬x n ) m = µ(1) = 1 and µ(¬yn ) k = µ(1) = 1. Thus µ(x n → y) = µ(1) = µ(y n → x).…”