Some derivations on EQ-algebras

In this paper, first the notion of (⊗,)−𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑜𝑛𝑠 on 𝐸𝑄−𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠 where ∈{~,→} is introduced and then they are characterized, in this way, the function 𝑑:𝐸→𝐸 on 𝐸𝑄−𝑎𝑙𝑔𝑒𝑏𝑟𝑎 𝐸 with the bottom element 0 is a (⊗,~)((⊗,→))−𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑜𝑛 if and only if 𝑑(𝑥)=1 ,𝑥~𝑦=1, for all 𝑥,𝑦∈𝐸. Thus 𝑑 is an isotone, homomorphism, and closure derivation. Furthermore, (⊗,∧)−𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑜𝑛𝑠 on 𝐸𝑄−𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠 are investigated and some related results are discussed. Indeed it is shown that if 𝑑:𝐸→𝐸 be a (⊗,∧)−𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑜𝑛 on an 𝐸𝑄−𝑎𝑙𝑔𝑒𝑏𝑟𝑎 𝐸 such that for all 𝑥∈𝐸, 𝑑(1)⨂𝑥≤𝑑(𝑥), so 𝑑 is an isotone derivation. Then by studying on (⊗,∨)−𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑜𝑛𝑠 on 𝑙𝑎𝑡𝑡𝑖𝑐𝑒−𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝐸𝑄−𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠, semi-dual closure derivation is introduced and it is proved that for 𝑎∈𝐸, 𝑑𝑎(𝑥)=𝑥⨂𝑎, for all 𝑥∈𝐸, is a semi-dual closure (⊗,∨)−𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑜𝑛 on 𝐸. Finally, by proving a property of 𝐸𝑄−𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠, some conditions are found under which the set of fixed points in (⊗,∨)−𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑜𝑛𝑠 on 𝑙𝑎𝑡𝑡𝑖𝑐𝑒−𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝐸𝑄−𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠, also is a 𝑙𝑎𝑡𝑡𝑖𝑐𝑒−𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝐸𝑄−𝑎𝑙𝑔𝑒𝑏𝑟a.