Show that two sets of equations define the same subvariety of semigroups

Let E_1 = \{x = xyx\} and E_2 = \{x^2=x, xy = xzy\}. Show that these sets of equations define the same varieties of semigroups. (That is, if S is a semigroup, then the equations in E_1 hold for all elements if and only if the equations in E_2 hold for all elements.)


Suppose S is a semigroup such that for all x,y \in S, x = xyx. Let x,y,z \in S. Now x = xxx and x = xx^2x, so that x = xxxx = xx. So x = x^2 for all x. Moreover, xy = (xy)(xy) = x(yx)y = x(yxzyx)z = xzy for all x,y,z.

Now suppose S is a semigroup such that for all x,y,z \in S, x = x^2 and xy = xzy. In particular, for all x,y \in S, we have x = xx = xyx.

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