<p>We introduce 3-hypergraph semigroups and 3-hypergraph semirings built from 3-hypergraphs <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation> and study the varieties they generate. We show that all 3-hypergraph semirings <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(S_{\scriptscriptstyle \mathbb {H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mstyle displaystyle="false" scriptlevel="2"> <mi mathvariant="double-struck">H</mi> </mstyle> </msub> </math></EquationSource> </InlineEquation> are nonfinitely based and subdirectly irreducible. Also, we prove that each variety generated by 3-hypergraph semirings is equal to a variety generated by 3-uniform hypergraph semirings. It is well known that both the variety <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textbf{V}(S_c(abc))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">V</mi> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (see J. Algebra 611: 211–245, 2022 and J. Algebra 623: 64–85, 2023) and the variety <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textbf{V}(S_{\scriptscriptstyle \mathbb {H}_3})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">V</mi> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mstyle displaystyle="false" scriptlevel="2"> <msub> <mi mathvariant="double-struck">H</mi> <mn>3</mn> </msub> </mstyle> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where the 3-uniform hypergraph <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {H}_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">H</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> is a 3-cycle, play a key role in the theory of varieties of ai-semirings. We show that each variety generated by 2-robustly strong 3-colorable 3-uniform hypergraph semirings is equal to the variety <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\textbf{V}(S_c(abc))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">V</mi> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and each variety generated by so-called beam-type hypergraph semirings or fan-type hypergraph semirings is equal to the variety <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\textbf{V}(S_{\scriptscriptstyle \mathbb {H}_3})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">V</mi> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mstyle displaystyle="false" scriptlevel="2"> <msub> <mi mathvariant="double-struck">H</mi> <mn>3</mn> </msub> </mstyle> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Finally, an infinite ascending chain is provided in the lattice of subvarieties of the variety generated by all 3-uniform hypergraph semirings. This implies that the variety generated by all 3-uniform hypergraph semirings has infinitely many subvarieties.</p>