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Abstract In this paper, we study the following max-type system of difference equations of higher order: { x n = max { A , y n − t x n − s } , y n = max { B , x n − t y n − s } , n ∈ { 0 , 1 , 2 , … } , $$ \textstyle\begin{cases} x_{n} = \max \{A ,\frac{y_{n-t}}{x_{n-s}} \}, \\ y_{n} = \max \{B ,\frac{x_{n-t}}{y_{n-s}} \},\end{cases}\displaystyle \quad n\in \{0,1,2,\ldots \}, $$ where A , B ∈ ( 0 , + ∞ ) $A,B\in (0, +\infty )$ , t , s ∈ { 1 , 2 , … } $t,s\in \{1,2,\ldots \}$ with gcd ( s , t ) = 1 $\gcd (s,t)=1$ , the initial values x − d , y − d , x − d + 1 , y − d + 1 , … , x − 1 , y − 1 ∈ ( 0 , + ∞ ) $x_{-d},y_{-d},x_{-d+1},y_{-d+1}, \ldots , x_{-1}, y_{-1}\in (0,+ \infty )$ and d = max { t , s } $d=\max \{t,s\}$ .