New-from-old full dualities via axiomatisation

We study different full dualities based on the same finite algebra. Our main theorem gives conditions on two different alter egos of a finite algebra under which, if one yields a full duality, then the other does too. We use this theorem to obtain a better understanding of several important examples from the theory of natural dualities. We also clarify what it means for two full dualities based on the same finite algebra to be different. Throughout the paper, a fundamental role is played by the universal Horn theory of the dual categories.