On the Strong Atoms of Q-Algebras

The concept of strong atoms in Q-algebras is discussed herein. In this work, some properties of strong atoms are provided. We show that there is no Q-algebra X with |X|≥3 such that all elements are strong atoms. We also find that any two-element subset of X containing a constant 0 is a subalgebra of X whenever X contains a strong atom. Moreover, any subset of X with the cardinality equal to 3 containing a strong atom and a constant 0 is always a subalgebra. We present some results concerning the concept of an ideal. In a Q-algebra X that contains a strong atom, any ideal of X is a subalgebra of X. An ideal of a Q-algebra X that is induced by any subset containing a strong atom is equal to X. Furthermore, we show that, for any Q-algebra X with a strong atom, there is only one ideal containing a strong atom. In particular, for |X|≤4, we propose that a finite union of ideals of X is again an ideal of X.