Question 1: Does every structure $\mathcal{A}$ have an $\mathcal{A}$-minimal degree?

No. Diamondstone, Greenberg, and Turetsky show that the set of array noncomputable degrees is a degree spectrum.

And Downey, Jockusch and Stob showed that a degree is array noncomputable iff it computes a pb-generic set. Now if $G=A\oplus B$ is pb-generic, then so are $A$ and $B$, and they are strictly Turing below $A\oplus B$.

Question 2: Is there a linear order $\mathcal{L}$ which has a $\mathcal{L}$-minimal degree?

Yes. Russell Miller showed that there is a linear order whose degree spectrum includes every nonzero $\Delta^0_2$ degree, but not $\mathbf 0$.

And Sacks showed that there is a minimal Turing degree that is $\Delta^0_2$; so we are done.

#References

David Diamondstone, Noam Greenberg, and Daniel Turetsky, Natural large degree spectra, Computability 2 (2013), no. 1, 1--8.

Russell Miller, The $\Delta^0_2$-spectrum of a linear order, J. Symbolic Logic 66 (2001), no. 2, 470--486.

Question 1: Does every structure $\mathcal{A}$ have an $\mathcal{A}$-minimal degree?

No. Diamondstone, Greenberg, and Turetsky show that the set of array noncomputable degrees is a degree spectrum.

And Downey, Jockusch and Stob showed that a degree is array noncomputable iff it computes a pb-generic set. Now if $G=A\oplus B$ is pb-generic, then so are $A$ and $B$, and they are strictly Turing below $A\oplus B$.

Question 2: Is there a linear order $\mathcal{L}$ which has a $\mathcal{L}$-minimal degree?

Yes. Russell Miller showed that there is a linear order whose degree spectrum includes every nonzero $\Delta^0_2$ degree, but not $\mathbf 0$.

And Sacks showed that there is a minimal Turing degree that is $\Delta^0_2$; so we are done.

References

David Diamondstone, Noam Greenberg, and Daniel Turetsky, Natural large degree spectra, Computability 2 (2013), no. 1, 1--8.

Russell Miller, The $\Delta^0_2$-spectrum of a linear order, J. Symbolic Logic 66 (2001), no. 2, 470--486.

Question 1: Does every structure $\mathcal{A}$ have an $\mathcal{A}$-minimal degree?

No. Diamondstone, Greenberg, and Turetsky show that the set of array noncomputable degrees is a degree spectrum.

And Downey, Jockusch and Stob showed that a degree is array noncomputable iff it computes a pb-generic set. Now if $G=A\oplus B$ is pb-generic, then so are $A$ and $B$, and they are strictly Turing below $A\oplus B$.

Question 2: Is there a linear order $\mathcal{L}$ which has a $\mathcal{L}$-minimal degree?

Yes. Russell Miller showed that there is a linear order whose degree spectrum includes every nonzero $\Delta^0_2$ degree, but not $\mathbf 0$.

And Sacks showed that there is a minimal Turing degree that is $\Delta^0_2$; so we are done.

Question 1: Does every structure $\mathcal{A}$ have an $\mathcal{A}$-minimal degree?

No. Diamondstone, Greenberg, and Turetsky show that the set of array noncomputable degrees is a degree spectrum.

And Downey, Jockusch and Stob showed that a degree is array noncomputable iff it computes a pb-generic set. Now if $G=A\oplus B$ is pb-generic, then so are $A$ and $B$, and they are strictly Turing below $A\oplus B$.

Question 2: Is there a linear order $\mathcal{L}$ which has a $\mathcal{L}$-minimal degree?

Yes. Russell Miller showed that there is a linear order whose degree spectrum includes every nonzero $\Delta^0_2$ degree, but not $\mathbf 0$.

And Sacks showed that there is a minimal Turing degree that is $\Delta^0_2$; so we are done.

#References

David Diamondstone, Noam Greenberg, and Daniel Turetsky, Natural large degree spectra, Computability 2 (2013), no. 1, 1--8.

Russell Miller, The $\Delta^0_2$-spectrum of a linear order, J. Symbolic Logic 66 (2001), no. 2, 470--486.

Question 1: Does every structure $\mathcal{A}$ have an $\mathcal{A}$-minimal degree?

No. Diamondstone, Greenberg, and Turetsky show that the set of array noncomputable degrees is a degree spectrum.

And Downey, Jockusch and Stob showed that a degree is array noncomputable iff it computes a pb-generic set. Now if $G=A\oplus B$ is pb-generic, then so are $A$ and $B$, and they are strictly Turing below $A\oplus B$.

Question 1: Does every structure $\mathcal{A}$ have an $\mathcal{A}$-minimal degree?

No. Diamondstone, Greenberg, and Turetsky show that the set of array noncomputable degrees is a degree spectrum.

And Downey, Jockusch and Stob showed that a degree is array noncomputable iff it computes a pb-generic set. Now if $G=A\oplus B$ is pb-generic, then so are $A$ and $B$, and they are strictly Turing below $A\oplus B$.

Question 2: Is there a linear order $\mathcal{L}$ which has a $\mathcal{L}$-minimal degree?

Yes. Russell Miller showed that there is a linear order whose degree spectrum includes every nonzero $\Delta^0_2$ degree, but not $\mathbf 0$.

And Sacks showed that there is a minimal Turing degree that is $\Delta^0_2$; so we are done.