An Equi-join is a join where the condition (predicate) is an equality.
All other join operations with a predicate other than equality are called theta join
An equi-join links two relations (tables, …) on common values in a column in relation 1 with a column in relation 2.
The restriction conforms to the following syntax when the condition is specified:
<MATH> \LARGE R \bowtie_{R.A = S.A} S </MATH>
Table1.column_a = Table2.column_a
A natural join is a shorthand equi-join where the equality applied to all shared attributes.
<MATH> \LARGE R \bowtie_{attrs(R) \cap attrs(S)} S </MATH> where:
<MATH> \LARGE R_1 \bowtie_{ a = b } R_2 = \sigma_{a=b} ( R_1 \times R_2 ) </MATH>
The join of R1 and R2 on A=B is the same that the selection of the cross product of R1 and R2 where A=B.
See this article : SQL - (Equi|Simple|Inner|Natural) Join